Here is the full list of all 25 Nickelodeon All-Star Brawl trophies - there are 6 bronze, 11 silver, 7 gold and 1 platinum.
Collect all the trophies.
Clear Arcade with all (standard) brawlers
Clear Arcade with 15 brawlers
Complete 50 online quickplay matches
Complete 10 online quickplay matches
Defeat an opponent with a higher rating than you 5 times in online mode
Defeat an opponent without losing a single stock in an online battle
Clear Arcade with 10 brawlers
Clear Arcade with 5 brawlers
Complete an Arcade run without losing a single stock
Defeat two level 9 CPUs in a team-battle where you are alone on your team
Defeat three level 9 CPUs in a team-battle
In Sports, win a game with yarn ball vs CPUs with a 10 score lead
In Sports, win a game with plankton ball vs CPUs with a 10 score lead
In Sports, win a game with soccer ball vs CPUs with a 10 score lead
In Sports, win a game with football vs CPUs with a 10 score lead
In Sports, win a game with Tommy ball vs CPUs with a 10 score lead
Defeat a level 9 CPU in a 1v1 battle
Play 50 local battles
Play at least once with all standard brawlers in a local battle
Compete in 10 local battles
Complete an online quickplay match
Complete an Arcade run on the highest difficulty
Clear Arcade with 1 brawler
Compete in your first local battle