These moves may be attempted anytime the Special Meter is flashing yellow. It should be noted that just because the meter is flashing does not mean you have the speed or balance necessary to perform these tricks.
Because this game was released on three different platforms, it is easier to list the type of move rather than what button to press.
For grab tricks: Use Circle on the Playstation, C-Right on the Nintendo 64, and B on the Dreamcast.
For flip tricks: Use Square on the Playstation, C-Left on the Nintendo 64, and X on the Dreamcast.
for grind tricks: Use Triangle on the Playstation, C-Up on the Nintendo 64, and Y on the Dreamcast.
Bob Burnquist
Backflip: Up, Down, Grab
Burntwist: Left, Up, Grind
One-Footed Smith: Right, Right, Grind
Kareem Campbell
Frontflip: Down, Up, Grab
Kickflip Underflip: Left, Right, Flip
Casper Slide: Up, Down, Grind
Private Carrera
The Well Hardflip: Right, Left, Flip
Somi Spin: Left, Right, Grab
Ho Ho Ho: Left, Up, Grind
Officer Dick
Yeehaw Frontflip: Down, Up, Grab
Assume the Position: Left, Left, Grab
Neckbreak Grind: Left, Right, Grind
Rune Glifberg
Kickflip McTwist: Right, Right, Grab
Christ Air: Left, Right, Grab
Front-Back Kickflip: Up, Down, Flip
Tony Hawk
540 Board Varial: Left, Left, Grab
360 Flip to Mute: Down, Right, Grab
Kickflip McTwist: Right, Right, Grab
The 900: Right, Down, Grab
Bucky Lasek
Varial Heelflip Judo: Down, Up, Flip
Fingerflip Airwalk: Left, Right, Grab
Kickflip McTwist: Right, Right, Grab
Chad Muska
Frontflip: Down, Up, Grab
360 Shove It Rewind: Right, Right, Flip
One-Footed 5-0 Thumpin': Right, Down, Grind
Andrew Reynolds
Backflip: Up, Down, Grab
Triple Heelflip: Left, Left, Flip
Heelflip to Bluntslide: Down, Down, Grind
Geoff Rowley
Backflip: Up, Down, Grab
Double Hardflip: Right, Down, Flip
Darkslide: Left, Right, Grind
Elissa Steamer
Backflip: Up, Down, Grab
Judo Madonna: Left, Down, Flip
Primo Grind: Left, Left, Grind
Jamie Thomas
Frontflip: Down, Up, Grab
540 Flip: Left, Down, Flip
One-Footed Nosegrind: Up, Up, Grind
Source: http://www.planettonyhawk.com/moves/thps/
I hope nobody minds that I consolidated these. Just wanted to tidy things up a bit (and add the info for alternate platforms). If it's a problem, let me know.