I usually have a few types of preferences.
1. Arcade: Car handling is outright wacky, but is nuanced in specific ways. Saints Row 2 is one of the better examples of this, but Just Cause 2 is probably the best example.
2. Unrealistic but gameplay-balanced: Self-explanatory. Halo 1, Quake 3 Rally, Interstate '76, and GTA3 are generally this. Vin Diesel: Wheelman is probably the best example.
3. Semi-realistic: Driver: the Gran Turismo series, Mafia 2 (Simulator handling mode), and GTA4.
4. Uberrealistic: Simulators.
If car handling in a game isn't true to one of those four types, I usually can't get into it too much. GTA5 for some reason isn't nuanced enough for me, and Saints Row 3's vehicles are fairly boring. The worst examples I can think of are that one Godfather game (where vehicles are a combination of squirrely and bizarre) and The Saboteur (where the vehicles are beyond boring).