99 lives:

Hold L2 + R2 on controller two and press Up, Down, Left, Right at the level selection screen. Then hold L1 + R1 on controller two, and press Right, Left, Down. Finally, release L1 and R1, then press Up.
All damage takes 1 point:

Hold L1 + R1 on controller two and press Triangle, X, Square, Circle at the level selection screen. Then, hold L2 + R2 on controller two and press Circle, Square, X. Finally, release L2 and R2, then press Triangle.
Fuwafuwa Potato:

Pause game play and press R1, L2, R2, L1, Left, Right, Circle on controller two.
Invincibility in caves:

Press R2(2), L2(3), R1(7), L2(3), R2(3), R1(7)on controller two at the level selection screen.
Ice Potato:

Pause game play and press R1, L2, R2, L1, Left, Right, Square on controller two.
Jumping Potato:

Pause game play and press R1, L2, R2, L1, Left, Right, Triangle on controller two.
Marine Potato:

Pause game play and press R1, L2, R2, L1, Left, Right, X on controller two. Padapada Potato:
Pause game play and press R1, L2, R2, L1, Left, Right, Down on controller two.
Refill energy:

Pause game play and press Up, Up, Down, Down, Square, Circle, Square, Circle, L1, R1 on controller two.
Rocket Potato:

Pause game play and press R1, L2, R2, L1, Left, Right, Up on controller two.
Unlock all levels:

Hold Circle + X + Triangle + Square + L1 + L2 + R1 + R2 on controller two and press enter Down, Down, Right at the level selection screen.