This will work for both Nascar '98 and the Collector's Edition of the game but only on the original Sony Playstation, not the Sega Saturn.
Hidden Pinnacle Trading Cards Car:
Highlight Bobby Labonte's car in the car selection part of Single Race, then hold X and tap Up and Down quickly.
Hidden EA car:
Highlight Kenny Wallace's car in the car selection part of Single Race, then hold X and tap Up and Down quickly.
Turbo Mode:
At the game options screen, hold Circle and press Up, Left, Down, Right.
If done correctly, you'll hear the pit crew saying go and a new option will appear at the bottom of the screen.
This cheat will make the game's speed extremely fast.
Paintball Mode:
To activate paintball mode, start a race and then pause the game.
After that go to "Race Statistics" and press L1 + L2 + R1 + R2 all at the same time, you'll hear an engine sound if done correctly.
Hold Triangle to shoot paintballs, which will spin out your opponents.
Note that the color of the paintballs is the same as your car's mini-map blip.
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Timestamps:
Pinnacle Trading Cards Car 0:00
EA Car 0:18
Turbo Mode 0:33
Paintball Mode 1:09