Level select:
R1, L1, Circle, Square, L1, R1, Triangle, Select.

All ships:
L1, Square, L1, Square, R1, R1, Circle, Select..

Invincibility:
L1, Circle, L1, Square, R1, R1, Circle, Select.

All medals and ranks:
L1, R1, L1, Circle, R1, Triangle, L1, Select.

Kobayashi Maru difficulty setting:
L1, Triangle, L1, L1, Square, Triangle, R1, Select.

Thanks to Revolution reader Troy Kaz!