Background Information!
Mew is popularly known as
The Phantom Pokemon for it's illusive mysterious nature throughout the series and is quite a fitting title as Mew is still one of the few event legendaries that is actually really hard to get as of USUM which unleashed legendary mayhem to the public, at least until Let's Go's take on the Mew Event. It's less popular but actual Pokemon category title however is
The New Species Pokemon. It's category title comes partly from the fact that at one time, it was a secret Pokemon even unknown to Nintendo at the time. By the time magazines revealed this, people assumed Mew was at the out-of-bounds truck at Vermillion City. According to it's Pokedex Entry, Mew is made up of the genetic composition of all Pokemon, a myth-turned-fact that's explored in Pokemon Adventures Emerald Arc when the soil of Mew's birthplace, Faraway Island, is used to calm almost all 386 Pokemon (the total of Pokemon up to Gen 3). One of Mew's most impactful roles is easily the very first chapter of Pokemon Adventures, where Red naively challenges the cute but absurdly strong legendary only for his very first Pokemon to get one-shotted, giving Red a harsh realization that he still has a lot to learn about Pokemon. However it's most famous debut started in Mewtwo Strikes Back where it battled it's cloned counterpart, Mewtwo, near the end of the movie. Anyways, enough about the background behind Mew.
Phantom Mew (Defensive/Staller Set)
Mew
Item: Leftovers
Ability: Synchronize
Nature: Calm (+SpD/-Atk)/Lax (+Def/-SpA)/Bold (+Def/-Atk)/Careful (+SpD/-SpA)
EVs: 252 HP/236 Def/16 SpD or 252 HP/236 SpD/16 Def
Taunt/Double Team/Stealth Rocks/Heal Bell/Trick (Choice Band over Leftovers)
Toxic/Will-o-wisp
Roost/Explosion
Psychic/Psyshock/Ice Beam/Knock Off/Foul Play/Explosion
As the Mew's more famous title implies, this is supposed to be a very annoying Mew set to deal with, one with heals, tankiness and stalling power even against the best stallers. The problem with this Mew set is that Mew has so many great moves but it's combos are limited to it's four move slots. Another problem is its outdated stats. Mew is no longer that special in competitive today due to the fact that it is a Jack-of-all-trades but master of none, having access to all TM/Tutor moves but is greatly outclassed by much more powerful tankier and trickier sets such as Cresselia or Skarmory who function the trickers/tank role far better than Mew can be. This doesn't mean Mew is bad, just outclassed in many ways. Offensively it has to count on STAB but it can count on Foul Play for decent damage against high physical attackers like Garchomp. If using Choice Band, one might consider having Trick on Mew to lock certain Pokemon moves. And like any decent tank set, what fun is to toy with people without considering the almighty annoying power that is Double Team, Taunt, Status Moves and recovery moves. However if one does not think that Mew will not be sufficient at recovery, one might want to consider the move Explosion as a last resort
New Species Mew (Offensive/Baton Passer)
Mew
Item: Mewnium Z
Ability: Synchronize
Nature: Timid (+Spe,-Atk)
EVs: 252 SpA/252 Spe
Psychic
Focus Blast/Aura Sphere/Nasty Plot
Rock Polish
Ice Beam/Flamethrower/Ancient Power/Baton Pass
As the name implies, this set intend to use the full force of what Mew has to offer or pass on it's power to it's descendants, the other Pokemon Team Members. Mewnium Z is a massive upgrade to a Pokemon that has been getting powercreeped by more and more legendaries that have mastered specific areas turning Mew into a Jack-of-all-trades but master of none. Essentially Mewnium Z is both an offensive and a setup move, delivering a 180-STAB attack against any non-Dark type while also setting up Psychic terrain which protects Mew from priority moves and gives Mew a massive boost in it's Psychic moves. For coverage, one may want to consider Ice Beam against common Dragon threas, Ancient Power on Volcarona or Flamethrower against common steel and bug-types. Rock Polish to increase Mew's sweeping potential through it's speed, allowing it to outpace other potential sweepers like Garchomp or Greninja. Some other moves that people may want to consider, due to the potential of how predictable Mewnium Z can be, is the old fashion Nasty Plot-Rock Polish-and-Baton Pass combo. With the Mewnium buff, this makes Mew a lot more intimidating but also somewhat predictable. Using this intimidation, one could bluff the opponent who might switch into a Dark-type for a Nasty-Plot or Rock-Polish and baton pass to a different Pokemon. Again, the problem with Mew sets in general is four move-slot syndrome as you have to decide between coverage moves and sweeper-setup potential.