Point by point:
Fully Prehensile Tails: if they are dextrous enough to use weapon in a fight, then they are, for all practical purposes, an extra arm - or rather, a tentacle. For this to work, the tail has to be quite long though. If it is not that dextrous, but is long enough, it can use to grapple/strangle an enemy. Might also help with Judo/Jiu-Jitsu moves, by giving extra grip to existing techniques and making new techniques possible.
Flat tails: two words: houndhouse tailwhip! Seriously now, you have an advantage that humans don't have - you can attack someone who is right behind you without having to rotate your torso for a back kick. It may also be a very strong, fierce attack.
Fluffy tails: if it is long enough, you can use it to distract your enemy by rubbing the tail against their face. Consider: such a tail might not do any physical harm in most situations, but you try and fight with a duster shoved against your face. The fur may also cause damage to eyes if it is hard enough.
According to east asian myths, foxes that have lived long enough will have an extra tail for each millenia they have lived. Such tails are connected to different kinds of magical or divine powers depending on the source you look up. If your world allows for magic, you might want to research about it.
A deer tail provides insignificant protection to your behind if you fall on it at best, and provides enemies with something to aim if it contrasts with the background at worst.
Stiff tails: another version of the flat tail. Effectiveness will depend on the freedom of movement the tail has. You can always used for a roundhouse tailwhip too.
Flashy or distracting tail (peacock's tail): you could maybe use bioluminescence - or, lacking it, some lights attached to your tail - to mesmerize or confuse your opponents. You can add a mix of taunts or awkwardness to enhance the effect. Bonus if you sing a song while doing it.
This is going to buy your allies a few precious seconds for them to disappear while your enemies beat you down.
Thin round tail (cats/great cats): these may be a little help in keeping balance, but this depends on the percentage of the body mass they represent. This should be much more effective for a small house cat than for a humanoid cat being.
Rattling tail (rattle snake): if you mean only the tip that rattles, it is no use in a fight. If you mean a snake-like tail, it works just like the prehensile tail from before.
One last, very important piece of info to add: while each of those tails could have different positive effects for their owners in a battle, they all have one single negative effect in common - it's one more limb for your opponents to use against you - by grabbing it, slashing it (causing you to lose blood), twisting it to immobilize you through pain etc. Your fictional creatures should be aware of this and include the necessary combat caution in whatever martial arts they develop.
Edit: To answer the following comment:
The only thing I was hoping someone would cover but no one did is if you had stiff tail that could support your weight, could you use it to momentarily balance on and kick your opponent?
As kangaroos do in comics, to kick with both feet at once. You could. This could probably work on a one on one fight, such as in a kick-boxing ring. In a brawl, though, that would provide your opponents with something to give a sweeping kick to from behind. Also such a tail is good in the open, but if you have to put your back to a wall, that might be the worst tail to have.