Here is the full list of all 24 Graceful Explosion Machine achievements.
First S Rank
Complete Magentos Prime
Destroy 30 bullets using the Energy Sword in a single level
First A Rank
Complete a phase without dropping the combo multiplier
Perform a 4x (or greater) combo that uses every weapon
Fire 100+ Missiles in a level
Get a 16x combo
All weapons active
Complete Tutorial 1
Complete Quartz Catacombia
Complete Scarlet Epsilon
Complete the game
Complete every WARP+ level
Replay a level and get an improved rank
First S+ Rank
S Rank an entire planet
S Rank all planets
Finish a level without taking damage (excluding tutorial levels)
Get a 32x combo
Get a 48x combo
Get 20 hits with a single Sniper Beam blast
Finish a level without flipping (excluding tutorial levels)
Die with less than 10% left on the Enemies Left meter