Here is the full list of all 11 River City Melee: Battle Royal Special trophies - there are 7 bronze, 3 silver and 1 gold.
You have played against all teams.
You have won against Munakata and Matsudo's (boss) teams.
You have cleared Story Mode.
You have won without using a special move.
You've engaged in online combat.
You've engaged in free battle.
You've used Body Check (triple combo) in rage as Hasebe of the Cheerleader Team.
You've damaged your opponents on a bike in Pier Stage.
You've turned the slot in Factory Stage.
You've scored the ball in Ground Stage.
You've destroyed the floor in Fight Destruction Stage.