Here is the full list of all 32 Bleed achievements.
Clear Arcade Mode on Normal (or higher).
Clear Arcade Mode on Hard (or higher).
Clear Arcade Mode on V. Hard.
Clear Arcade Mode on Easy (or higher).
Defeat the Rival with an S-Rank.
Defeat the Bubble Core with an S-Rank.
Win a Challenge Mode fight against three opponents.
Defeat the Laser Bots with an S-Rank.
Defeat Bunny Rockette with an S-Rank.
Reflect a bullet that was reflected by the Rival.
Recite some poetry atop the Hall of Heroes.
Defeat the Mantis Core with an S-Rank.
Defeat the Spider Core with an S-Rank.
Defeat Gibby & Stu with an S-Rank.
Defeat the Dragon Heart with an S-Rank.
Defeat Battalion with an S-Rank.
Defeat the Mirror Core with an S-Rank.
Clear Story Mode on Easy (or higher).
Clear Story Mode on Normal (or higher).
Clear Story Mode on Hard (or higher).
Clear Story Mode on V. Hard.
Defeat White Mk II with an S-Rank.
Defeat Guppy with an S-Rank.
Defeat the Chopper Core with an S-Rank.
Investigate the pit in Guppy's Manor.