I currently have 2 C6Z's and have also previously owned 2 C6 coupes. The '06 coupe I drove to 88k miles.
Issues I've had to fix over time in all 4:
- Auto trans shifter connector replacement (Coupe)
- Crankshaft Position Sensor (Z06)
- Crank Pulley bolt (Coupe)
- Stick shift lower assembly (Z06)
All in all, they've been pretty reliable. My '09Z has 38k miles and was bought with 26k miles 5 years ago. Other than the shifter, it's been pretty reliable.
I won't go into details on the '08Z and all of the stuff that broke over the years, but it's been racing since Day 1 and is about to be on its 3rd engine.