Mew

Type Tier
Psychic Uber
Statistics
Stat
HP
100
403
Atk
100
298
Def
100
298
Spc
100
298
Spe
100
298
Name Moveset

Swords Dancer

~ Swords Dance
~ Earthquake
~ Hyper Beam / Swift
~ Softboiled / Explosion

Swords Dance up, then hit your opponent with a boosted Hyper Beam, or in the case of Gengar, Earthquake. Softboiled when necessary, or self-KO if you have a threat you want to eliminate and you don't anticipate the ability to sweep. Swift is an option if Mew is in an evasion metagame or if your opponent is lowering your accuracy. This set is countered by Barrier, specifically Barrier Mewtwo.

Name Moveset

OHKO

~ Thunder Wave
~ Horn Drill
~ Fissure
~ Softboiled

Paralyze your opponent if it's faster, and proceed to use Horn Drill or Fissure while using Softboiled when necessary. Fairly self-explanatory. This set is countered by paralysis and a Pokémon that is faster than Mew, such as Mewtwo, as well as evasion.

Name Moveset

Special sweeper

~ Thunderbolt
~ Blizzard
~ Softboiled
~ Psychic

Use Thunderbolt or Blizzard repeatedly until the enemy is KOed. Does fair and quick damage versus Pokémon that aren't pure Psychic or Normal-types, but it's unpopular due to its inability to address many common threats. The only special set that could ever sweep or do well early-game is one that aims to freeze an opponent's Pokémon, and only in a metagame that bans evasion; most of the time, this set will at best stall.

Other Options

Body Slam can be used to deal damage and paralyze other Pokemon. Thunder Wave is an option if you want a reliable paralysis option. If evasion is allowed, there are not many Pokemon that can use it better than Mew, but otherwise, you can go with Flash. Transform can also be used. Counter could be fun for Snorlax and Tauros. Fire Blast is a possibility for the power and Burn factor, but anything Fire Blast covers Ice Beam covers well enough (specifically, Exeggutor). Reflect, together with Rest, makes Mew a great tank. It can also Seismic Toss for that guaranteed 100 damage or Submission to use against Chansey. It has even further special moves like Surf and Thunder. In short, Mew can do a lot, which is what makes him so dangerous.

Opinion

There is a set for every Pokémon out there. Until it is disabled, either by KO, Sleep, or Freeze, it can and will wreak havoc on any Pokémon, given the appropriate set. In short, there is no standard defense against Mew. Without good Special as well as solid Defense, Mew will sweep. Swords Dance Mew remains the most effective and widely used set overall, however.

Counters

Mewtwo is better, and will beat Mew most of the time, but it depends on the set since Mew can go Swords Dance+Explosion on it; still, although uncommon, Mewtwo could have Barrier to stop that (plus Thunder Wave, Ice Beam, Amnesia/Rest/Recover). Putting Mew to sleep is a great idea because it's too versatile to really have a consistent counter. Likewise, trying to freeze it is an ideal situation since it would then be worthless.