Here is the full list of all 20 Explodemon trophies - there are 20 bronze.
Beat the par times for all levels
Destroy 15 enemies in less than five seconds
Beat a par time for any level
S-Rank all Squall levels
Completely upgrade Explodemon and unlock all of his abilities
S-Rank all Arbor levels
S-Rank all Nibia levels
Destroy 1000 Vortex
Complete Adventure Mode
Defeat Absorbemon without taking any damage
Bring down a Vortex dropship
Destroy 100 enemies by dunking them in acid
Close Explode the same green sticky 10 times
Destroy 10 enemies in less than five seconds
Perform a 3-Hit Combo
Complete any level with an A-Rank or higher
Explode 10 times in the air without touching the ground
Perform a Counter
Destroy five enemies in less than five seconds
Level up to Max Level