Here is the full list of all 49 BULLET SOUL INFINITE BURST achievements.
Cleared stage 4 in burst mode
Cleared stage 3 in burst mode
Collide against an enemy during a burst
Cleared stage 2 in burst mode
Used 2 bursts in a stage
Used a burst
Cleared stage 1 in burst mode
Beat a boss in a moment during a burst.
Recovered HP
Cleared a game in caravan mode
Missed in caravan mode
Play a game to get in Piece of Cake
Play a game to get in Red Hot Skull
Play a game to get in FA-98 Stinger
Play a game to get in Turtle 10000
Cleared stage 1 in normal mode
Cleared stage 2 in normal mode
Cleared stage 3 in normal mode
Cleared stage 4 in normal mode
Cleared stage 5 in normal mode
Cleared stage 5 in burst mode
Retried caravan mode
Break my highest score in caravan mode
Cleared a game without missing in caravan mode
Unlocked Musou mode
Clear stage 1 in Musou mode
Clear stage 2 in Musou mode
Clear stage 3 in Musou mode
Clear stage 4 in Musou mode
Clear stage 5 in Musou mode
Watched a ending with Zenichi
Watched a ending with Yun
Watched a ending with Sadahl
Wawtched the ending with loop
Cancel a burst manually
Kept a burst for 30 seconds
Earned 50 HP items with your HP full in total
Increased your coins' score rate up to 100,000 points
Earn 50 collection bonuses in total
Earned 25,000 coins in total
Earned Platinum Medal at stage 1 of Bancho mode
Earned Platinum Medal at stage 2 of Bancho mode
Earned Platinum Medal at stage 3 of Bancho mode
Earned Platinum Medal at stage 4 of Bancho mode
Earned Platinum Medal at stage 5 of Bancho mode
Earned Tamashii gauge 2,000 magnification in total
Earn 500 various bonuses in total.
Shot down 25,000 units in total
Earned 25 and over medals in total