There are a lot of formats that are only playable for a month at a time, so I don't know all of them. If you want a list of those, then just tell me and I'll hide my answer.
Singles
Random Battle: Players are given random (usually fully evolved) Pokemon with reasonably good movesets. The lower tier Pokemon come at higher levels.
Unrated Random Battle: Same as Randoms, but battles have no effect on rating.
Over Used: Pokemon, moves, and abilities can be banned by Smogon's tiering policy. The bans traditionally include most box art legendaries.
Under Used, Rarely Used, Never Used, and PU: Pokemon with more than 1-(1/2)^(1/15), or about 4.5%, usage in the above tier are banned, as well as some Pokemon that don't make the usage cutoff but are still deemed "broken".
Uber: All Pokemon species are allowed, and the bans on moves and abilities less restrictive.
Little Cup: The maximum level is 5. All evolved and fully evolved Pokemon, as well as the ones deemed "broken", are banned.
Anything Goes: Everything that can be achieved without cheating is allowed. The only exception is endless battles. In gen 8, this format also allows all Pokemon, moves, and items that exist in gen 7.
Create-a-Pokemon: Everything allowed in OU, as well as stuff made by Smogon's Create-a-Pokemon project, is allowed.
Battle Spot Singles: Each player enters up to 6 Pokemon, looks at the opponent's 6 Pokemon, and chooses 3 to use in battle. All Pokemon above level 50 will be set to level 50 for the battle. Box art legendaries and event-exclusive stuff are banned.
Doubles
Random Double Battle: Players are given random (usually fully evolved) Pokemon with reasonably good movesets. The lower tier Pokemon come at higher levels.
Doubles Over Used: Pokemon, moves, and abilities deemed "broken" are banned. The bans traditionally include most box art legendaries.
Doubles Under Used: Pokemon with more than 1-(1/2)^(1/20), or about 3.4%, usage in OU are banned, as well as some Pokemon that don't make the usage cutoff but are still deemed "broken".
Doubles Uber: Almost all Pokemon are allowed, and the bans on moves and abilities less restrictive.
Battle Spot Doubles: Each player enters up to 6 Pokemon, looks at the opponent's 6 Pokemon, and chooses 4 to use in battle. All Pokemon above level 50 will be set to level 50 for the battle. Box art legendaries and event-exclusive stuff are banned.
Video Game Championships: Each player enters up to 6 Pokemon, looks at the opponent's 6 Pokemon, and chooses 4 to use in battle. All Pokemon above level 50 will be set to level 50 for the battle. All Pokemon must be on the Alolan Pokedex. Legendaries and event-exclusive stuff are banned.
Other Metagames with permanent ladders (all of these are Smogon formats, so they all ban "broken" stuff)
Balanced Hackmons: Pokemon can have almost any moves and ability, and can be mega evolved without having a mega stone or having to mega evolve.
1v1: Each player enters up to 3 Pokemon, looks at the opponent's 3 Pokemon, and chooses one to use in battle.
Monotype: All Pokemon on each team must share at least one type.
Mix and Mega: Any Pokemon can mega evolve or primal revert using any mega stone or orb, and they gain the stat changes, abilities, and type changes associated with the mega stone or orb. The one mega evolution limit is lifted.
Almost Any Ability: Pokemon can have abilities that are normally illegal.
Sketchmons: Each Pokemon can know up to one move that's normally illegal.
Randomized Other Metagames
Battle Spot Singles Factory: Each player gets 6 random Pokemon with movesets that are viable in BSS.
Challenge Cup 1v1: Each player gets 6 random Pokemon with random but legal moves, abilities, and items, looks at the opponent's 6 Pokemon, and chooses one to use in battle.
Hackmons Cup: Each player gets 6 random Pokemon with completely random moves, abilities, and items.
Formats with no permanent ladders
Create-a-Pokemon Little Cup: Same as LC but the pre-evolved forms of Pokemon created by Smogon's Create-a-Pokemon project are also allowed.
Custom Game: Pokemon can have any moves and abilities, know up to 24 moves, and be up to level 9999.
Doubles Custom Game: Pokemon can have any moves and abilities, know up to 24 moves, and be up to level 9999.
Hidden Type: All Pokemon have an additional type that's determined by IVs, unless the type given by the IVs is the same as a type the Pokemon already has. Pokemon deemed "broken" are banned.
Balanced Hackmons Doubles: Pokemon can have almost any moves and ability, and can be mega evolved without having a mega stone or having to mega evolve.
Monotype Random Battle: Same as Randoms, but all Pokemon on each team share a type.
Doubles Hackmons Cup: Each player gets 6 random Pokemon with completely random moves, abilities, and items.
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