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