To unlock special cars, you should beat challenge sheets:
EA GT Racer: White Mountain's Challenge Sheet
Hot Rod: Lone Peak's Challenge Sheet
Low Rider: Angel Valley's Challenge Sheet
Nixon Special: Eastern Bay's Challenge Sheet
Etnies Racer: Motor City's Challenge Sheet
Euro Classic LM: Eternal City's Challenge Sheet
Custom Classic: Sunshine Keys' Challenge Sheet
Criterion GT Racer: Central Route's Challenge Sheet
Black Elite Racer: Gain the "Elite" Ranking
Revenge Racer: Gain "100%" Status