Big Head Mode:
Pause the game and press L1+L2+R2+Down simultaneously. You'll hear a sound to confirm.

All Sphere Mode:
Pause the game and press L1+L2+Circle+Square. You'll hear a sound to confirm.

Level Select:
Pause the game and press R1+Square simultaneously, then Triangle, Circle, X. You'll hear a sound to confirm.

Invincible:
Pause the game and simultaneously press O+Square+Right, then simultaneously press R1+R2+Left. You'll hear a sound to confirm.

Thanks to Revolution readers Adam MacNutt and Eddie Vonderbargenharg!