Zero mini-game:
Beat the game.

Copy-X mini-game:
Beat the game on hard.

Fefnir mini-game:
Beat every mission with 100 points.

Leviathan mini-game:
Beat every mission with 100 points.

Harpuia mini-game:
Beat the game with only the Zet Zaber and "S" rankings.

Ciel mini-game:
Beat the game with an "S" in every mission.

Phantom mini-game:
Beat the game with only the Buster Shot and "S" rankings.

X mini-game:
Beat the game, then use the same saved file to beat the game again.

Hard difficulty setting:
Beat the game, then highlight "New Game," hold L and press Start.

Ultimate difficulty setting:
Beat the game after collecting all the hidden disks. Then highlight "New Game," hold R and press Start.

Thanks to Revolution readers Dim Something, and Erika!