There are 20 Trophies that can be collected.

Bronze

Trigger Finger
Perform a Counter

Dangerous To Be Around
Level up to Max Level

The Way It Should Be Played
Complete any level with an A-Rank or higher

Save The Universe!
Complete Adventure Mode

Nibia Domination
S-Rank all Nibia levels

Arbor Domination
S-Rank all Arbor levels

Squall Domination
S-Rank all Squall levels

Look, 'Mon, I Can Fly!
Explode 10 times in the air without touching the ground

Speed Runner
Beat a par time for any level

Super Potential 1000% !!!
Completely upgrade Explodemon and unlock all of his abilities

Speed Runner 1000% !!!
Beat the par times for all levels

Quintuple Frag!
Destroy five enemies in less than five seconds

Decuple Frag!
Destroy 10 enemies in less than five seconds

Duocuple Frag!
Destroy 15 enemies in less than five seconds

Bring it down! Bring it down!
Bring down a Vortex dropship

Sizzle Sauce
Destroy 100 enemies by dunking them in acid

Street Fighter
Perform a 3 Hit Combo

Sticky Ball
Close Explode the same green sticky 10 times

Untouchable
Defeat Absorbemon without taking any damage

Weapon of Mass Destruction
Destroy 1000 Vortex

http://www.ps3trophies.com/forums/trophy-news/53987-explodemon-trophy-list.html