Here is the full list of all 25 Rym 9000 trophies - there are 7 bronze, 11 silver, 6 gold and 1 platinum.
Master the secrets of Rym 9000
Kill all Popcorn enemies in Area 1
Kill all Spirals in Area 1
Kill all enemies in the first half of Area 1
Don't receive any damage from Bombers in Area 1
Don't receive any damage from Area 1's boss
Finish Area 1
Kill all Popcorn enemies in Area 2
Kill all Detonating Shooters in Area 2
Kill all 3 Rangers in Area 2 with X3 multipliers
Don't get trapped inside a Ringer in Area 2
Defeat the 2nd phase of Area 2's boss in less than 10 seconds
Don't receive any damage from Area 2's boss
Kill all Popcorn enemies in Area 3
Kill all Targeters in Area 3
Don't receive any damage from Area 3's boss
Finish Area 3
Kill all Popcorn enemies in Area 4
Kill all 3 Crushers in Area 4 with X3 multipliers
Don't touch the screen borders in Area 4
Don't receive any damage from Area 4's boss
Don't receive any damage from Area 5's boss
Finish the game
Absorb all 8 gunner powerups in one run
Finish the game without absorbing any gunner powerups