I'd say the "best" handling cars have got to be:
The
'62 Skyline Sport Coupe and the
'54 Vette. No other cars in the game can you come blazing into a corner, brake when you're right on *top* of the corner and smoothly glide around. The Skyline is especially fun under hard turning because the weight shift is so dramatic the whole screen must tilt about 30 degrees
Other than that... there are just so many cars that I admire the handling of across the entire spectrum of handling styles. Here's some:
Toyota RSC - my go to car for NA Sports. Wonderful car to toss from side to side.
Impreza 22B-STi - both exceptional grip and controllable power slides at the same time!
Elise Type 72 and 111R - ohh... but how tight we corner!
Renault Clio V6 24V - A bit delicate but so, so smooth.
BMW M3 GTR,CSL and M5 - Now, do I really need to explain why these are on the list?
TVR V8S and
TVR Griffith - So much fun. Exceptional speed and just enough oversteer to be fun without being out of control.
Opera S2000 - A "rails" car.
Merc 190E - THE "rails" car.
Lotus Europa - Old school's idea of smooth
And then there's the fun "seat of your pants" kind of cars
Lancia Stratos - Unbelievable speed, Unbelievable cornering ability (although the line between gripping and sliding a corner is razor thin), Unbelievably difficult to tame. The Wild Mustang of GT4. If you want truly

driving... take this car into the Nurb 4H and leave it stock. YES, it can win although it's best to avoid the Speedster, NSXes, Elise and Garaiya.
Alpine A310 - You have to learn an all new driving style to tame this beast.
And that's just the tip of the iceberg.