I think this is all the cars in the game.

SLY JOKER: Beat the Pawn, Knight and Bishop tracks on regular
difficulty.

FALCON MK-II: Beat Pawn OR Knight OR Bishop tracks on Expert difficulty.

FIGHTING COMET: Beat the Pawn, Knight, Bishop and Queen tracks on Master
difficulty.

SILVER THUNDER: Beat the Queen track on the Expert difficulty.

STINGRAY: Beat the Pawn, Knight and Bishop tracks on the Expert
difficulty.

JET VERMILION: Unlock all other cars, then complete Championship mode
255 times. OR complete each series on Master Class with each of the
racers OR use your password.