Unless otherwise specified, all codes are entered at the pause screen. At any time while playing, pause the game and then enter the code. In most cases, the screen will shake to indicate the code has been entered correctly.
Playstation Codes
Slow Motion Mode: Hold L1 and press Square, Left, Up, Square, Left.
Stats to 10: Hold L1 and press Square, Triangle, Up, Down.
Stats to 13: Hold L1 and press X, Square, Square, Triangle, Up, Down. (Ten is the normal maximum for stats.)
Special Meter Always Full: Hold L1 and press X, Triangle, Circle, Down, Up, Right.
Skip to Restart: Hold L1 and press Square, Circle, X, Up, Down. (This allows you to start from different locations in each stage.)
Unlock All Levels: Hold L1 and press Triangle, Right, Up, Square, Triangle, Left, Up, Square, Triangle. (The levels will still appear unavailable but they can be selected anyway.)
Unlock Everything: Hold L1 and press Circle, Right, Up, Down, Circle, Right, Up, Square, Triangle.
Officer Dick: Obtain all 30 tapes in career mode with any character.
Private Carerra: Start a game in any mode using Officer Dick, then press pause. Hold L1 and press Triangle, Up, Triangle, Up, Circle, Up, Left, Triangle. The screen will not shake. Quit the game - it may freeze if you try to keep playing - and begin a new one. Private Carerra will replace Officer Dick.
Nintendo 64 Codes
Points Multiplied by Ten: Hold L and press Down, Right, Up, Right, Up, Left, C-Left.
Perfect Balance: Hold L and press C-Up, C-Right, Left, C-Right, Right, Up, Down.
Stats to 10: Hold L and press C-Left, C-Up, Up, Down.
Stats to 13: Hold L and press C-Down, C-Left, C-Left, C-Up, Up, Down.
Faster Grab Tricks: Hold L and press C-Up, Left, D-Down, C-Down, Up, Down, Right.
Slow Motion Mode: Hold L and press C-Left, Left, Up, C-Left, Left.
Turbo Mode: Hold L and press Right, Up, Down, Down, Up, Down.
Skip to Restart: Hold L and press C-Left, C-Right, C-Down, Up, Down.
Earn all Tapes: Hold L and press C-Right, Left, Up, C-Up, C-Up, Right, Down, Up.
Bonus Picture: Hold L and press C-Left, C-Down, C-Right, C-Left, C-Down, C-Right, C-Right. (A picture of a woman appears on the pause screen; she is believed to be Tony Hawk's wife Erin.)
Officer Dick: Obtain all 30 tapes in career mode with any character.
Private Carerra: Start a game in any mode using Officer Dick, then press pause. Hold L and press C-Up, Up, C-Up, Up, C-Right, Up, Left, C-Up. The screen will not shake. Quit the game - it may freeze if you try to keep playing - and begin a new one. Private Carerra will replace Officer Dick.
Dreamcast Codes
Slow Motion Mode: Hold L and press X, Left, Up, X, Left.
Special Meter Always Full: Hold L and press A, Y, B, Down, Up, Right.
Stats to 10: Hold L and press X, Y, Up, Down.
Skip to Restart: Hold L and press X, B, A, Up, Down.
Unlock All Levels: Hold L and press Y, Right, Up, X, Y, Left, Up, X, Y.
Unlock Everything: Hold L and press B, Right, Up, Down, B, Right, Up, X, Y.
Officer Dick: Obtain all 30 tapes in career mode with any character.
Private Carerra: Start a game in any mode using Officer Dick, then press pause. Hold L and press Y, Up, Y, Up, B, Up, Left, Y. The screen will not shake. Quit the game - it may freeze if you try to keep playing - and begin a new one. Private Carerra will replace Officer Dick.
Gameshark Codes
There are a lot of Gameshark codes for all four platforms. To keep this reasonable, I'm only including some of the more unusual ones. If you must have all of the codes for all of the characters for all of the platforms, consult the sources section below.
Big Head Mode
PSX: 800D1724 0001
DC: 1E841689 00000001
Ether Mode
PSX: 800D1722 0001
DC: 1D7C1689 00000001
Infinite Time
PSX: 80025556 2400
Paper Thin Skaters
PSX: 8008B6A2 2400
Skate Upside Down
PSX: 80085182 2400
Trick Tips Modifier
N64: 8029E293 00??
Score Display Modifier
N64: 8029E292 00??
Infinite Time on Halfpipe
GBC: 0104D6C4
Infinite Powerups
GBC: 012C84C5
Infinite Speed Coins
GBC: 0109D6C4
Faster Turbo Boosts
GBC: 01103EC3
Sources
http://www.planettonyhawk.com/codes/thps/
http://www.planettonyhawk.com/gscodes/thps/
I hope nobody minds that I consolidated these. Just wanted to tidy things up a bit, and to add the information for alternate platforms and Gameshark. If it's a problem, let me know.