Here is the full list of all 22 Ultracore trophies - there are 14 silver, 7 gold and 1 platinum.
Collect all coins in any level
Collect all lives in any level
Complete level 5
Get 60% hit accuracy in any level
Complete level 4
Don't touch the ground for 30 seconds
Complete level 3
Use all weapons
Complete level 2
Buy something from the vending machine
Kill 95% of enemies in any level
Uncover a secret area (by bombing it)
Complete level 1
Upgrade standard weapon to maximum level
Uncover a secret wall (by walking into it)
Use a bomb
Collect all trophies
Complete the game
Complete the game using just the standard weapon
Kill 95% of enemies in the entire game
Collect all lives in the entire game
Collect all coins in the entire game