Shadow Claw + Dragon Claw and Wild Charge

Likely the most common "generic" moveset in Great League, this set gives solid coverage over many of the format's top threats.

Shadow Claw + Dark Pulse and Overheat

In Ultra League, a format defined by Registeel, Cresselia, and Giratina, this set gives you the best shot at beating all 3. Do note that you will struggle against Giratina no matter what, though, given its bulk and Ghost-type damage.

Shadow Claw + Ice Beam and Focus Blast

For Master League, consider this set! Ice Beam gives you great coverage against Groudon and all the Dragons, and Focus Blast lets you nuke down Steel-types!

Other Options

In the end, Mew’s greatest strength is in its versatility, so selecting from canned movesets loses a large part of what makes it useful. As such, any enterprising Mew trainer is recommended to take a close look at their team, and use Mew to fill any remaining holes. Or just spend all the charged TMs you have and see what you get.

As a rule, you can never go wrong by starting out your set with Shadow Claw. The move's incredible neutral coverage, solid damage, and great energy gain allows it to effectively leverage its fantastic charged move pool. Volt Switch is better when you need that Electric-type damage (especially against Azumarill or Skarmory in Great League!). Snarl, on the other hand, gives up a lot of damage for a bit of energy gain, which can definitely matter in some situations!

Once you've got your fast move of choice, pair that with any combination of these Charged Moves: