One vs many ships
First off, I would separate transport into three parts: surface-to-orbit (= earth to space), orbit-to-orbit (space travel from earth to titan), and orbit-to-surface (= landing on titan). For all three parts, I would suggest different vessels because the requirements are so radically different.
Since we haven't improved any on the "Strap capsule with transport load to a rocket" design for surface-to-orbit travel in the past 60 years of space travel, I doubt we will be able to come up with a radically different design within the next 20 years. I am saying 20 years because leaving you 40 years for going from a scientific design idea to a workable prototype to a final efficient form (that passes all kinds of safety regulations when it's intended for mass transport) and then mass-producing it in a large enough numbers to evacuate earth, is very short.
Surface-to-orbit will, by necessity, be a smaller vehicle where you can transport only a couple hundred to a couple thousand people at the very most -- it needs to be somewhat aerodynamic, after all, and capable of escaping the gravity well. Also, there comes a point where the fuel-per-person ration increases for every additional passenger and makes large rocket-style shuttles inefficient. (Reason: the fuel for the additional person also weighs something, so you've got to stow some more fuel to transport the additional-person-fuel until it is needed).
Orbit to orbit
The orbit-to-orbit vessls can be huge super-ships that can transport a percentage of humanity (several hundred thousand to several million). You can also include all kinds of amenities and necessities like gravity for the inhabitants (a rotating ring) that make a surface landing completely impossible.
From a scientific stand-point, there might be a size limit due to the overheating problem. Yes, space is cold, but there is no better insulator than vacuum. And pretty much everything on a space ship produces heat -- from the humans on board to the engines to all computers and appliances. You do bleed off heat 'normally' through infra red radiation, and you can increase cooling by using all kinds of decompression cooling (i.e. let miniscule amounts of water or other liquids 'evaporate' into space). However, both ways are limited. Radiation cooling is proportional to surface, but heat is proportional to volume (i.e. heat production growth quicker than cooling capabilities when you scale up). And for evaporation cooling you need to carry something with you that you can evaporate -- at a certain point that becomes inefficient.
There also might be a couple other factors that don't scale well beyond a certain point; fuel once again bein amongst them, same for material strength (you wouldn't want your engine to take off with only half the ship because the I-beams holding everything together snapped).
A more compelling argument against one super-ship would be politics. Since I doubt that by 2080 we will have a unified world government, you will probably never be able to convince all countries to work together on one single space craft. Especially not when anything beyond simple scientific interest is on the line. Also consider that you might not want to have warring factions on the same space ship -- too dangerous that war erupts and then destroys/damages the ship. And you will have to deal with politics and command-lines on the ship, too -- keeping hundreds of thousands of people living together peacefully in a sardine can is very, very hard work.
And last but not least -- the most prominent argument against one supership has already been mentioned by the other posters: laying all eggs into one basket is never advisable.
Orbit to surface
Once again, this will have to be smaller vessels because the orbit-to-orbit cruiser can't land on a surface. And the landers have to be transported by the orbit-to-orbit cruiser.
General feasibility of evacuating earth
I doubt you will ever manage to evacuate the entire earth population. Aside from technical limitations (can we produce enough spaceships to provide a spot for every human? can we stem the transport volume required to get everyone off earth within a reasonable amount of time? can we produce enough fuel to get the spaceships off the ground? can we make the spaceships autark enough that whole cities can survive in there for years?), you have human nature as a limitating factor.
- whoever produces the spaceships will not want to let everyone on board (criminals? people who can't pay the transportation fee? radical elements? ideology / religion?)
- there will be non-believers (the world will end in 20 years? tell that to your grandmother!)
- there will be people who refuse to leave earth
- there will be a technical failure somtime down the line that kills an entire shuttle of people -- less people will be willing to leave after that
- there will be people who can't survive the stress of take-off, let alone such a long space voyage
- there will be countries / populations / religions that deliberately go against what the majority says (just because)
- once it becomes clear that there are definite deadlines, panic will explode and either kill a lot of humans or damage important surface-to-orbit transport infrastructure
I'd say getting a billion (out of 10 to 15 billion in 2080) willing and capable of leaving earth is a very, very generous estimate. Even that would require transporting 300,000 people off planet per day for 10 years despite riots and technical failures.
And in space -- it took the Cassini Huygens seven years to reach Titan (https://en.wikipedia.org/wiki/Cassini%E2%80%93Huygens). If you want to have a chance at ferrying 1 billion people across to Titan, you should probably start right now.
Actually, I'd suggest Mars as an evacuation target -- temperature is reasonable when compared to Titan, you get more sunlight as cheap energy source, it is closer to earth, its gravity is closer to earth, and no atmosphere is about as helpful as a poisonous super-cold nitrogen atmosphere when trying to settle somewhere. On Mars you have at least a chance at warming the planet with the green-house effect; on Titan you can forget that.
Individual vs. Mass transport ships:
Single-person-ships will probably not be feasible due to the human factor -- families don't want to be separated, and since the journey will be measured in years you'd have to deal with the psychological torture of isolation. Even if you put them on ice for the journey, I doubt that people would send off their small children and babies like that.
Family- or group-sized crafts (10-100 people): While it might be technically feasible to mass-produce peresonal surface-to-orbit transportation, who would buy one since it's a one-way journey? I'd expect something like a regular shuttle service, and that will be optimized to get as many people into orbit as possible.
Small-sized orbit-to-orbit transportation is even more unlikely since it takes years to get there. There is a lot of knowledge and on-the-fly troubleshooting necessary for such long voyages since we don't have the routine and experience yet. Also, I would expect the ration of loaded-to-empty to scale very well up until you hit the upper boundary of technical feasibility. Just imagine cargo ships today -- bigger means more efficient. And if you are talking about the insane amount of resources for an exodus, efficiency becomes the new religion.
The only option left are large to huge orbit-to-orbit crafts. How people get on and off however is up for discussion.