Mew

You have to complete seven of the Special Research tasks in order to encounter and capture Mew (which I believe is the 8th and final task of the Special Research), so it's not random. Getting to that point is no easy task, though.

I definitely don't have Mew yet, as I just got to the second task earlier today, so I've got a long way to go to reach the little mythical creature.

I caught mine today. The quest was a lot of work but it was fun, too. I'm looking forward to more.

The catch mechanics are different with Mew than other Pokemon. When you claim your final reward (Mew), you go directly to the catch screen. At least on the iPhone, AR mode was on, and I could not turn it off. You are also restricted to using only normal balls (although the supply seems to be unlimited?) and no berries. When you touch the ball, Mew becomes invisible save only for its outline.

So providing you can actually hit it (it is, after all invisible and jumps around a lot), it has a 100% catch rate. On every one I've seen, including mine, it pops out of 3 balls before finally staying.

Just finish the special quest and it will appear. It's actually quite easy, the hardest part is to find a Ditto and 10 ghost types. Got it yesterday, as many people did.
You'll get unlimited Pokéballs and no berries, and Mew has a 0% flee rate.

If you are looking for the research tasks:

https://www.reddit.com/r/TheSilphRoad/comments/88g3j4/how_to_catch_mew_step_by_step_credits/

We can only get one mew, right? So, there's no reason to *not* power him up, even if not ideal?

I still have a while to get there, but just curious for whenever I do.