At the 'Press Start' screen, enter the following. 'Press Start' will flash if you did it correctly.

Invincibility:
Right, X, R1, Triangle, Up, L2, X, Left, L1, Circle, X, R2 .

All weapons:
Up, Down, X, Square, R1, Triangle, Triangle, Left, Circle, L1, L1, Right.

Level select:
R2, Triangle, Left, Circle, Square, L2, Left, Up, X, Down, L2, Square.

Full bolt:
Left, Right, Up, Down, L1, Circle, Triangle, R2, Left, Down, Square, Square.

Full spread:
L2, R1, Circle, L2, Down, Up, L1, Right, Left, X.

Full homing:
Right, Up, Square, L1, Left, Left, L1, Left, Circle, Left.

Full beam:
Left, Circle, Triangle, Right, L1, Square, Left, R1, R1, Triangle.

Full area effect:
Left, X, Triangle, Up, X, Down, Square, L2, Left, R2.

No power-ups:
Down, Up, X, Square, Down, Up, X, Square, L1, L2, Square, Circle.

Training missions:
Square, Up, L2, Left, Triangle, X, R2, Circle, Right, R1, Square, Circle.

Boss mode:
R1, Triangle, Down, Down, X, L2, Left, Square, Right, Triangle, R2, L1.

Agent data:
Up, Down, Circle, R2, Left, L2, Right, X, R2, Square, Up, R1.

Alien data:
Square, L1, Circle, L2, Down, Triangle, R1, Right, X, Left, R2, Triangle .

"Making Of" Feature:
Circle, R2, L2, Circle, Triangle, Down, Square, X, Right, L1, X, Up .

Thanks to Revolution readers Andrew, Chris T, bradleymhines and Marcus Boyer!