So I've seen this question as to how a cape could work as a shield, but why else would 'classic' superheroes (Batman, Superman) wear a cape? Is there something about a cape that makes a person more heroic or does it have a practical purpose?
In classic comics, the cape provides the ability to add drama to the hero's pose, often physically filling the frame to emotionally make the hero the centre of attention and even convey a sense of movement when the character is standing still (the cape flowing in the wind).
Capes are also part of the costume of nobles and aristocracy from the middle ages into the 1700's. Comic book artists educated in the early part of the 20th century and their audiences would be well aware of this, so there is a subtle signature of the aristocratic nature of the character. (As an aside, cartoons like Bugs Bunny filmed in the period between the 1940's and early 1960's often expected viewers to get jokes based on classical opera and other fairly complex subjects, which should give you an idea of what educated people used to be like).
In the period from the 1500's to the 1700's, cloaks actually did have a practical purpose (besides protecting the wearer from weather) as being secondary weapons or improvised shields for men fighting with the sort of rapiers or small swords which were common in that time period.
While very few superheroes fight with or are attacked by swords these days, the use of cloaks as improvised weapons and shields is referenced by the occasional use of a cloak as an improvised shield by comic book heroes.
NO CAPES The capes conversation in Incredibles captures it best.
Don't get me wrong, I've always been an apologist for Batman's cape. It's awesome and I convinced myself it's bulletproof or a glider, or useful for intimidation or whatever. But if you need bullet protection wear a bulletproof vest. If you need a glider, arrange for a glider where you need it. And honestly who can be intimidated by a grown man in a cape?
Capes are great at getting snagged on things, billowing when you don't want, making noise, did I mention getting caught on things?
If you really think it through you'll realize that they're more of a liability than an asset.
Imagine a ninja: death from the darkness, skill woven into every fiber, stealth and agility through the roof. The most deadly assassin ever to stalk the earth. Put him in a cape and suddenly he's ridiculous. No capes.
In Superman's case, fashion matters. He's an alien whose powers could easily make most people scared of him. He needed to win people over by looking heroic and patriotic. His mother did a great PR job in picking that iconic cape and S.
Batman uses his for camouflage. And for psych terror effect (he really does look like a giant bat). As noted in the comments, it is also viable as a glider.
From the artistic side, the point of the cape is simple. It's the same reason wrestlers tend to have long hair. The cape is an action line that is used to accentuate the action that is taking place. This was needed more in the old days due to printing quality, but as time went on it wasn't needed so people stopped designing characters with capes. That's why the majority of Marvel characters don't have them. They were designed in the 60s and after where as many of the DC heroes were designed in the 30s.
As far as whether they are practical on a character, depends on the character. On Superman the idea was that it acts as an airfoil that helps control his flight, but this was back when he was either transitioning to having flight or didn't have flight. Artistically it also serves as a show of damage. Superman and his costume doesn't get hurt, but his cape gets ripped up as he fights.
Batman's cape is used for gliding, misdirection, to hide his body. make him look bigger and like a bat, as a way to shift some of the protection from his suit to the cape (for example the cape is electrically heated), it serves as a way to extend protection to others, and lastly as a weapon with bladed ends on the scalops. Robin uses his cape for much the same reasons, but Robin I's cape was used to extend his shape and cause criminals to fire at it rather than him. Robin III+ use it as a means of increased stealth with its black outside.
In the Supergirl TV series, when Winn is making Supergirl's costume, he omits the cape at first, and then adds it when he realizes the drag helps her to control her flight.
Many heroes use their capes as an impromptu bag or pouch to carry children, animals, or other small objects. Particularly Superman; see Action Comics Vol 2 13, where he does this to an injured Krypto. In Batman: The Animated Series, Bats used his cape to scoop up water to try to put out a fire. And tries to capture Pig Diana in the same way in Justice League.
For heroes who can't fly, the cape provides drag when falling.
In combat, the cape provides a form of protection. It is gives the hero a large profile, most of which is actually empty space. A hit to the cape might look solid, but actually do nothing.
Cape can be used as an impromptu pad or blanket.
An armored cape can trap a sword. Or possibly slow bullets.
I'm sure the whole purpose of a cape is to look cool, especially in dramatic poses in comic books.
Spawn had a living cape. It looked awesome. The artist (Todd Mcfarlane?) could fill whole splash pagees with the curls and whorls of Spawn's cape with Spawn perched dramatically in the middle. This was probably the pinnacle of the visual drama of the cape. After such excess it's hard to do anything new and interesting with a cape. This sort of thing sometimes inspires an artistic backlash (that's overdone, I'm doing something else) It's been a while though so maybe it's time for a retro refresh of the aesthetic.
The main purpose of a cape is for psychological effect. It makes the wearer look bigger, more noticeable, more "apparent", which is good for a superhero that wants to represent a paragon of justice or to intimidate their enemies. Since that's basically the point of becoming a named and costumed superhero (as opposed to just being a regular crime fighter with powers) capes make sense - although they are offset by the impracticality of catching on things or getting in the way.
They can also be used as a shield and can be used to entangle an enemy's weapons (some swordfighters used capes for this purpose), or provide a bit of extra protection against bullets, especially if the cape is made of some sort of high-tech super-fabric.
In a fight, especially in the dark, a hero can also use the cape to break up their silhouette and confuse enemies into shooting the cape instead of their body. In a pinch, it can be thrown as a decoy as the hero makes their escape, or deployed as chaff to confuse the targeting system of homing missiles.
Watchmen touches on this idea.
I experimented with a cloak, remembering how the Shadow would use his cloak to misguide enemy bullets, leading them to shoot at parts of the swirling black mass where his body didn't happen to be. In practice, however, I found it unwieldy; I was always tripping over it or getting it caught in things, and so I abandoned it for an outfit that was as streamlined as I could make it.
Dollar Bill was one of the nicest and most straightforward men I have ever met, and the fact that he died so tragically young is something that still upsets me whenever I think about it. While attempting to stop a raid upon one of his employer's banks, his cloak became entangled in the bank's revolving door and he was shot dead at point-blank range before he could free it. Designers employed by the bank had designed his costume for maximum publicity appeal. If he'd designed it himself he might have left out that damned stupid cloak and still be alive today.
Though this is kind of insightful on one potential in-universe use of a cape/cloak, there are a lot of statements in that quote, some on the impracticality of the outfit design in-universe, others on problems with superheroes as a concept and still more on real-world corporate exploitation.
On the one hand, one guy used it effectively, another guy found it laughably impractical, a third guy's used it solely for the aesthetic and it got him killed. Three different characters that all handled it differently, and a different "moral to the story" for each character.
So really the design of your outfits is up to you, and depends on what messages you want to convey to your audience.
It can be very dangerous to wear a cape. You might get trapped in a revolving door.
PR and dramatic looks
Even if they keep their real identities secret, their hero identities are celebrities nonetheless. They must maintain image close to public's definition of "heroes", not "some uncontrolled militia with superpowers".
Also, good loks and proper PR pays. Can you imagine just how many starving children can be saved with proceeds from selling some advertising space on Superman's costume? In situation, where heroes compete for ratings you're not only going to see them wearing capes - they will do dialogues with villains mid-fight.
A lot of super hero costume choice is about trying to look cool. There are maybe practical benefits to looking cool, but do remember these choices come from comic books where image is everything.
Flags look pretty cool with a little wind. And a cape is like always having a flag with you and you don't even have to bother waving it if you keep moving.
It might also be noted some serious effort is being put into design of uniforms of soldiers and police, but capes are not part of the standard kit yet.
Capes are used for visual intimidation.
While some uses can be defined for a cape, it can be a cloak to guard against the elements and possibly some forms of attack, a cape mostly just makes wearer seems bigger, stronger, more imposing on his audience. This is the exact reason royalty and aristocracy used and, in some cases, still use them.
Does Superman need a cape for protection? He's the man of steel. He just looks awesome with it. Without a cape, he's a dude in tight, long underwear.
Does Batman need a cape? Granted, he does use it for protection sometimes and possibly gliding but, mostly, he just looks awesome and scary with it. And being a "bat" man, he needs something that resembles black, leathery wings.
So, there are quite a few answers that deal with the negatives of capes. I had a few other ideas, but whether or not a cape is useful or practical depends on whether the person using it can balance those negatives and positives.
A cape might block visual lines - not just breaking up a silhouette or using its motion in combat to draw fire away from the body (though those are also useful), but in general there might be any number of scenarios where it can be positioned or used to literally make it hard to see what the person wearing it is doing - hiding someone's hand while reaching for a weapon or tool, letting them pick something up or put it down (or tuck it in a pouch or pocket) without being seen, letting them hook their foot or brace their weight - as long as their shoulders aren't moving, it's hard to see what might be moving when the line-of-sight is physically blocked by the cape.
Additionally, the person might use the cape, and their own profile, to physically block someone seeing past them (blocking their view of something or someone else). They might use the cape to conceal something bulky or noticeable that they're carrying, or just anything they'd rather not be seen with. They might hide wounds, or wear-and-tear, behind the capes - especially if the cape is used for misdirection, it might be expected to come off the worse for wear, and so even in rough shape it might not betray it's wearer's actual damage to the casual eye. Most superhero uniforms are form-fitting (for ease of movement, among other things) - so if there's anything, anything they don't want being instantly visible, they need some way to conceal it - and a loose fall of fabric will do so easily, if they should just happen to have one on hand. Of course, it isn't foolproof - a line-of-sight might not be blocked by the cape, or they may not be allowed to conceal something without being challenged, depending on the positions of people involved and the actual scenario - but having a cape means it's an option.
A cape might have armor or all kinds of protection - fireproof, bulletproof, etc. Of course, there are uniforms that have those kinds of protections built in, that will give better protection and better ease of movement to the person wearing them - but what something like a bulletproof vest won't do, is let someone offer that protection to someone else, or even more than one, at least not without a lot of shenanigans (undressing and whatnot). So the superhero can wrap their cape around someone else a lot quicker and easier than trading off a tailored uniform vest - and it will give a lot more coverage that just a vest would, or even the superhero wrapping themselves around the second person.
A thicker, heavier armored cape (not the thin cosmetic kind, okay) might provide additional protection by forming a kind of, makeshift crumple zone - a blow not only has to get through the cape, but also through the layer of air between the cape and the body, and then get through the protection on the armor. A tightly fitting armor, even one with layers, will transmit kinetic force through it and into the person wearing it much more efficiently than one with a gap between layers, especially one where the outer layer crumples into the lower (spending energy dragging the whole cape inwards) and fluffs itself right back out afterwards. Some kinds of blows will get the cushioning much more than others - something narrowly focused and sharply penetrating might not deform the outer layer (not using the 'crumple zone') and so the only variable is the total thickness of the armor and not whether there's space between layers, while something broad and blunt-force might lose a lot of energy being dispersed along the cape.
And then there are ways a cape might be modified to a specific superhero's use - a cape weighted at the hem (for better draping) might make a kind of cloth club, or even a bola - and what is weighting that hem might itself be useful. A minimum of added framework or other modification might make a kind of glider, to slow down and control falls. It might be given some property that works with the superhero's abilities, or against a usual villain's - being able to conduct, or resist, some energy or typical combat conditions - with a bonus of being able to hold that property at a distance or quickly drop it if some scenario turns that property against them (which would be a lot harder if it was built into the uniform). It can just have all the extra pockets, since none of the heroes ever seem to need more space to carry stuff in. It maybe can be used for visual signalling (if said hero tends to work with others, and semaphore code is useful).
Of course, there are the historical reasons for a cape or cloak to consider - shelter from the weather, keeping warm or dry or cool and shaded, a makeshift blanket or pad, or a bag or net to catch or carry with, or even providing fabric to tear into makeshift ties or bandages. A thinner, less armored version of a cape might be better for these uses (or, perhaps if there's a non-armored cloth lining that might be torn out). Coats are more often used for weather proofing in modern times, but a cloak or cape might have multiple uses - not the least of which is the ability (which a tailored coat may lack) to extend that protection to others. It may also be easier to adjust a cloak or cape to different temperatures - anywhere from tightly wrapped to keep warmth in to loosely draped to let heat flow to propped up to create shade, its wear can be adjusted a lot more than a more tailored coat. The simpler construction (a big rectangle, more or less) should also make it easier to re-purpose for other uses, like a blanket, or a bag, or a pad - tailoring which makes it a better coat, for example, will also make it harder to use for other purposes - and if it must be torn up for bandages or ties, a more closely tailored coat represents more lost work (to remake or replace) than a simply cut rectangular cape lining. How often your superhero might need the cape for these purposes is unknown, but the multi-purpose capability might be something they thought of when designing their costume.
Really, the ability to wrap a cape around someone else seems to be a really strong benefit - your superhero only has to grab an edge of the cape and swing it around someone to protect them from fire or bullets, keep them dry or warm in poor weather, or even serve as a shock blanket - as long as it is cut large and loosely draped enough for coverage. It's a lot easier than carrying around separate protection (especially since it's easier to throw around someone in a hurry carried loose on the back, instead of packed away) - and by draping it around themselves, it can serve as an extra layer of protection when they're not using it for someone else. That might be enough for your superhero to justify having a cape even if other uses (concealment, misdirection, emergency source of cloth, makeshift shield) aren't enough to counterbalance the negatives.
Have you ever seen runners being wrapped in "space blankets" at the end of a race? Someone who's just done lots of heavy physical activity will be sweating, but now they've stopped there's a risk of their temperature dropping.
Similarly, superheroism is exothermic. The hero or heroine's lycra outfit is great for shedding heat while they're working, but when they stop it can become distinctly chilly. Especially at high altitude. So they can wrap themselves in the cape to keep warm.