Here is the full list of all 22 Polybius trophies - there are 6 bronze, 8 silver, 7 gold and 1 platinum.
Get to level 10
Complete a level without losing a shield
Finish a level under full boost
Complete any level
Achieve ultimate transcendence
Get every fried egg in level 6
Get maximum shields in Pure mode
Try all three game modes
Get to level 13 in YOLO mode
Finish the game in Normal mode
Get to level 30 in Pure Mode
Finish a level without shooting
Do three consecutive levels without losing a shield
Get 10 'Perfect' accuracies in one game
Use 10 lift gates without landing
Fly for 50 seconds or longer
Become a bull
Hands off the controls for 30 seconds
Do 5 consecutive levels without losing a shield
Do 10 consecutive 1000+ point slalom gates
Use 5 consecutive Boost pills with no collisions
Kill 1000 cattle during a game