Pause the game and enter the following:

Big feet:
Right, Right, Right, Left, Right, Left, Right, Left, Left, Left.

Big gun:
X, X, X, X, Y, Y.

Skeleton:
Left, Left, Left, Right, Left, Right, Left, Right, Right, Right.

Explosive crossbow:
Left, Right, Right, Left, X, Y, R, L, X, Y, Left, Right.

Explosive sniper:
R, L, Right, Right, Left, Left, Right, Right, L, R, X, Y.

Missile crossbow:
Right, Right, Left, Left, R, R, L, L, Y, Y, X, X.

Missile sniper:
X, Left, Right, R, L, Right, X, L, L, R, Left, Left.

Half damage:
X, X, Y, Y, Left, Left, Right, Right, Y, Y, X, X.

No bullet damage:
Left, Right, Right, Left, X, Y, Y, X.

One hit kills:
Y, Left, Right, Right, Left, Y, R, L, Y, X, X.

Unlimited bullets:
X, Y, X, Y, X, Y, X, Y, Left, Y, X, Y, X, Y, X, Y, X.

Unlimited health:
Left, Right, Right, Left, Left, Right, Right, Left, Left, Right, Right, Left, X, X.

Unlimited Spy Cam:
Left, Left, Right, Right, L, L, R, R, X, X, Y, Y.

Unlimited TOC:
Y, Y, X, X, Left, Right, Right, Left, R, L, R.

Skeleton mode:
Left, Left, Left, Right, Left, Right, Left, Right, Right, Right.

Level skip:
R, X, R, Y, R, Left, R, Right, R, L, L, X, L, Y, L, Left, L, Right, X.

Level completion codes must be entered within the corresponding level to work.

Bank level completed:
L, R, X, Y, Left, L, Left, Left, X, Y, X.

Carmen level completed:
L, R, Right, Left, Left, R, L, X, X, X, L.

Forsythe level completed:
R, R, Right, Right, Right, R, L, X, L, X, L.

Installation K level completed:
L, R, X, X, Left, R, R, X, X, L1, X.

La Casa level completed:
L, R, Right, Left, Left, R, L, X, X, X, L.

Magyar level completed:
L, Y, Y, Left, Left, R, L, Right, X, Right, Right.

Mod level completed:
L, R, Right, L, Left, Y, L, Y, Right, Right, Left.

Museum level completed:
R, R, Right, Right, Right, R, L, X, L, X, L.

Reliance level completed:
L, R, Right, Left, L, R, Right, X, Y, X, Y.

Silo level completed:
L, R, Right, Left, Left, R, Y, L, X, X, X.

Training level completed:
Left, Right, Right, Left, Y, X, L, R, Y, Y, X.

Thanks to Revolution readers Da Shoza, Bill James, and Tiny P!