64/74 Shiki Mode:
At the title screen, press UP, Up, Down, Down, Right, Left, Right, Triangle.

Atlancer Mode:
At the title screen, press Left, Right, Right, Down, Left, Left, Right, Triangle.

Movie Version:
At the title screen press Down, Down, Left, Down, Down, Left, Down, Triangle.

Queen Fairy Mode:
At the title screen, press Left, Right, Left, Down, Down, Right, Right, Triangle.

Star Geppy X:
At the title screen, press Down, Left, Left, Down, down, Right, Right, Triangle.

Wild John Mode:
At the title screen, press Up, Down, Right, Up, Down, Right, Up, Triangle.

Thanks to Revolution reader Jason Sommerfeld!