Here is the full list of all 21 Jump Stars trophies - there are 6 bronze, 6 silver, 8 gold and 1 platinum.
Unlock all the Trophies in Jump Stars
Entered into Tournament gamemode
Entered into a custom gamemode with 1 game in the playlist
Entered into a custom gamemode with 9 games in the playlist
Punched 1,000 times overall in the game
Punched 5,000 times overall in the game
Punched 10,000 times overall in the game
Jumped 1,000 times overall in the game
Jumped 5,000 times overall in the game
Jumped 10,000 times overall in the game
Gained 25,000 teamscore overall in the game
Gained 50,000 teamscore overall in the game
Gained 100,000 teamscore overall in the game
Gained 250,000 teamscore overall in the game
Gained 1,000,000 teamscore overall in the game
Eaten 1,000 Pies in Collect overall in the game
Broken 1,000 barriers in Drop overall in the game
Achieved 3,000 teamscore in a mini-game in a single play
Completed Tournament (inlcuding Gauntlet) with 2 players
Completed Tournament (inlcuding Gauntlet) with 3 players
Completed Tournament (inlcuding Gauntlet) with 4 players