To enable the following codes you have to enter them at the 'Main' menu. If you've entered the code correctly a confirmation message will appear at the bottom of the screen.

Unlock Level Select:
Press X, Y, Z(2), X, Y, Z(2).

Unlock Invincibility:
Press L, R, L, R, L, R, L, R.

Infinite Hero Time:
Press B, X, Y, Z, L, R.

Unlock Exploding Bullets:
Press L, R, Z, Y, B.

Unlock Exploding Fists:
Press R(2), Y, B, X, R(2).

Unlock Hot Fists:
Press L(2), X, B, Y, L(2).

Unlock Liquid Metal Mode:
Press B, Y, X, B, Y, X.

Unlock Flame On Mode:
Press X, Y, B, X, Y, B.

Unlock Big Head Mode:
Press R(2), L, R.

Unlock Small Head Mode:
Press L(2), R, L.

Unlock Pin Head Mode:
Press B, X, Y, B, X, Y.

Unlock Hero Mode:
Press L + Y with a full Special Meter.

Exploding Bullets:
At the Main menu press the following combination to make your enemies explode with any shot: L, R, Z, Y, B

Full Body Explosion
At the Main menu press the following combination to learn the art of Full
Body Explosion: R, R, Y, B, X, R, R

Flame On
At the Main menu press the following combination to turn on Flame On:
X, Y, B, X, Y, B

Invincibility
At the Main menu press the following combination for Invincibility:
L, R, L, R, L, R, L, R, L

Liquid Metal Mode
At the Main menu press the following combination to get Liquid Metal mode: B, Y, X, B, Y, X

Unlock All Levels
At the Main menu press the following combination to unlock any level:
X, Y, Z, Z, X, Y, Z, Z

For the following codes, enter them at the main menu, a message will be displayed to confirm correct code entry

Level select
X, Y, Z(2), X, Y, Z(2)

Invincibility
L, R, L, R, L, R, L, R

Big heads
R(2), L, R

Small heads
L(2), R, L

Hot fists
L(2), X, B, Y, L(2)

Exploding bullets
L, R, Z, Y, B