Here is the full list of all 31 MIGHTY GUNVOLT BURST achievements.
Clear a stage without using any Bursts
Get a Burst Combo of 10+ & clear a stage
Clear the opening stage
Defeat a boss using Burst
Defeat all 8 of the Mighty Numbers
Clear the game as Beck
Clear the game as Gunvolt
Clear the game as Ekoro
Clear the game in under 1 hour
Beat all bosses using their weaknesses
Clear a stage without taking any damage
Clear a stage with an 'S' rank
Clear a stage with 1500+ CP of equipment
Clear all stages using Custom Slot No. 1
Clear the stage under certain conditions