Abnormalcy: Single battle, best of 3, standard counterpick. You are not allowed to use your A button. [Source: grbsh]
BBBest Tournament: Single battle, best of 3, standard counterpick. You are not allowed to use your A, C, or super moves. If your B move can be modified by a forbidden button, that's allowed. [Source: Jobert and Remy]
Best Three of Four: Team battle, best of 1. Each match, choose 4 characters - your opponent will remove one of these characters, leaving you with your final team. [Source: EmiSocks]
Chaotic: Team battle, best of 1. You must play with a ruleset for every match. The ruleset is chosen at random using a random selector - if the players do not agree on the ruleset selections they may reroll up to 3 times.
Deck: Single battle, best of 3, standard counterpick. Each character can only appear in one match in the tournament, including matches with custom movesets - if you run out of characters, you forfeit.
Democracy: Single battle, best of 3. At the start of the match, each player chooses their opponent's character, and the loser of a game may counterpick by changing their opponent's character.
Lawful: Team battle, best of 1. You must play with a ruleset for every match.
Roll the Dice: Single battle, best of 3, standard counterpick. At the start of the tournament, select six characters and number them from 1 to 6. Each match, roll a d6 (physical or digital) to select your character. The loser of a game can "counterpick" by repeating the process.
Mirror Mayhem: Single battle, best of 3, no counterpicks. Both players must pick the same character, chosen randomly from the cast. Only rulesets that do not contradict this character choice may be played. After losing a game, a player may counterpick by repeating the random selection process for both players.
The Many and the Few: A random number is picked from 1-5 at the start of each match. That is how many characters are in the participants' teams, using a team battle manager. If 1, simply do a 1v1 match. Best of 1, unless it's 1v1 in which case best of 3.
Play Fair: Single battle, best of 3, full character repick. At the start of every game, before character selection, both players are separately assigned a random Play Fair ruleset. They must then pick more Play Fair rulesets until they have 4+ points - with more than 4 points, they only need 3 points next game. Neither player should say their ruleset until after the game. [Source: Ritielko]
Simplify: The winners' bracket has teams of 3. The losers' bracket has teams of 2, using a team battle manager. Best of 1. [Double Elimination only] [Source: Danielizabeth Stretulch]
Teams: Single battle, best of 3, no counterpicks. All players are grouped into two teams, where they must each play from a given set of characters, chosen on a per-tournament basis. The teams play sequential matches against each other according to a custom team battle manager until one team wins.
You Are What You Beat: Single battle, best of 3, standard counterpick. Upon completing a game, the player that wins will change to their opponent's character for the next game. [Source: fuu buu]
Your Own Worst Enemy: Single battle, best of 3, no counterpick. Each player's starting character is chosen at random. Upon completing a game, the player that loses will change to their opponent's character for the next game, and the player that wins rerolls a new character. [Source: ErickDRedd]