Level select:
At the main menu, press Up, Left, Circle, Circle, Left, Left, Square, Down, Up, Circle.

Level skip:
During play, press Square, Triangle, Left, Down, Right, Triangle, Right, Up, X.

Invincibility:
During play, press X, Triangle, Right, Left, Circle, Up, Down, Down, Right, Circle, Circle.

Extra damage:
During play, press X, Triangle, Right, Up, Right, Triangle, Circle, Right.

Burn everything:
During play, press Square, Triangle, Left, Down, Square, Circle.

Goat mode:
During play, press Triangle, Right, Left, Left, Circle, Triangle, Square, Right, Up.

Thanks to Revolution readers Steven Landells and Adam Wilson!