Here is the full list of all 23 Gal*Gunvolt Burst trophies - there are 8 bronze, 7 silver, 7 gold and 1 platinum.
Clear a stage without using any Bursts
Defeat all 8 of the Mighty Numbers
Clear a stage with 4000+ CP of equipment
Clear a stage with 2000+ CP of equipment
Clear a stage with an "S" rank
Clear a stage with 1500+ CP of equipment
Defeat a boss using Burst
Get a Burst Combo of 10+ & clear a stage
Clear the opening stage
Obtain every other trophy
Clear the game as Beck
Clear the game as Gunvolt
Clear the game as Ekoro
Clear the game on Hard difficulty
Clear the game in under 1 hour
Beat all bosses using their weaknesses
Clear a stage without taking any damage
Get a Burst Combo of 30+ & clear a stage
Get a Burst Combo of 50+ & clear a stage
Clear a stage with 6000+ CP of equipment
Clear all stages using Custom Slot No. 1
Beat the final stage in under 10 minutes
Earn a score of 1000 during the credits