Well, spoilers are my friend here. Analysis of Mew's movepool, with viable options categorized. I'm not going to discuss bad options because I'd be here all week, so I'll just categorize what I think is worth it-
Alternatively, if you're looking for competitive moves it doesn't have, the list is the base 40 power physical priority moves, ExtremeSpeed, Mirror Coat, Confuse Ray, Spore, Spikes, Toxic Spikes, Rapid Spin, Quiver Dance, Dragon Dance, and Shell Smash.
It does everything
. Ever. Need a dual screens + kaboom lead? Mew. Need a Baton Passer? Mew. Want a good end game sweeper with decent bulk and speed? Mew. Need something that can take on fighting moves? Mew. Need some surprise factor? Mew. Need any niche that has ever existed? Mew. I can not begin to suggest sets that would work with it. Any walling, attacking, mixed wall/attacking, set-up, baton pass, or support set that has ever existed that doesn't include the 15-odd moves listed above, you can do it with Mew, and it should work out alright. There's not really a definitive set for it, and nothing can be argued to be the best, simply because it does everything ever. I once had a question of, "What has priority, phasing, and SR?" to which the answer was only Empoleon. Except now Mew also happens.
I suppose for a set to actually really qualify as a stand-out, it should be something that only Mew can do. Quite frankly, I'm inclined to give it a Baton Pass set, simply because I believe that's what it does best. However, seeing as Baton Pass relies on team support and is everyone's favorite strategy, I suppose I'll go with something else. Killing top tier threats is good, everyone likes momentum, and 100 base speed is definitely decent. So uh...
Mew @ Choice Scarf
EVs: 252 SpAtk / 252 Spd / 4 SDef
- Volt Switch
- Ice Beam
- Earth Power
So uh... yeah. Very specific moveset, if you couldn't tell. Darmanitan, Mienshao, Heracross, Raikou, Flygon, Crobat, Zapdos, and various other things that are scary kind of actually die to this. You're walled silly by Bronzong and Snorlax, but that's why Volt Switch exists. Ice Beam and Earth Power do a nice job of killing Electric and Ground types that may try to switch in on Volt Switch (except SpDef Rhyperior, which takes between 31-36%, so you need some damage on that first). Mew's natural bulk also allows it to take hits switching in, and it should be able to take two unboosted moves from pretty much anything as long as it isn't super effective. You will have a speed tie with Flygon/Zapdos (not that Zapdos runs one) if they're scarfed with speed-boosting nature, and Mienshao and Raikou outspeed as a scarfers, so be wary of them. Darmanitan, Crobat, Heracross, and non-scarfed anything other than Ninjask will not be so happy when they come in head-to-head against Mew and expect to outspeed. Oh, and Volt Switch exists. Momentum is always fun, not to mention that coverage you get on water types.
Also, oddly enough, this comes with perfect neutral coverage for everything in UU (screw Shedinja and all it stands for). The four types are just unrelated enough to the point that it can't be done. The only way to actually do it would be if Magnezone got Levitate, if I'm not mistaken. I found it kind of funny and odd, so figured I might as well share it.
EDIT: If I may, I'd like to request Empoleon for the next round. I feel like there's a lot there to touch on and a creative moveset is possible, and having it show up here will hopefully have people recognize it as a scary thing that's worth using.