At the Options screen, hold L1 + R1 and enter the following:

Invincibility:
Triangle, X, Triangle, X

Faster cars:
Square, Square, Square, Square

Much faster cars:
Triangle, Triangle, Triangle, Triangle

Instant explosion:
Triangle, Triangle, Square, Square

Super jumps:
Square, Square, Square, Triangle. Use the horn to jump.

Alternate view:
Circle, Circle, Circle, X

Show speedometer:
Triangle, Triangle, Circle, Square.

Grid view:
Circle, X, Circle, Triangle

Freakout view:
Triangle, Circle, Triangle, Circle

Itchy and Scratchy Cartoon:
Collect every card in the game and go to Lisa'a level. In the Android's Dungeon, talk to the Comic Book Guy and buy a ticket for the cartoon. Then, go watch the cartoon at the Aztec Theater.

New Year's Theme:
Set the system date to January 1st.

Halloween Theme:
Set the system date to October 31st.

Christmas Theme:
Set the system date to December 25.

Thanks to Revolution readers Butterman and Mark87!