Invincibility:
Press START at any time during the game to pause, then press R, Right, A, L, Right, A, Down, Y and unpause. Now your character is invincible. Stage Select:
Press START at any time during the game to pause, then press L, A, Z, Y, Left, Right, A, Down, Right, L. A cheat menu will appear, allowing you to switch characters, access a sound test, enable the "flying" cheat or skip to any stage, including the bonus rounds.

Thanks to Revolution reader Matthew Cervantes!!