Here are some of the most common defects that have been identified.  Most of these problems have been addressed by BMW in either a Service Bulletin or Factory Recall.

Frayed top material:  Material pulling loose around the rear window has been noticed in the entire 3 Series, apparently due to poor quality control at the subcontractor.  Solution is simple, have your dealer replace the top under warranty.

Erratic Fuel Gauge:  Fuel gauge problems as severe as the low fuel indicator illuminating and the gauge indicating empty immediately after fill-up.  Replacement of the fuel sending unit is required to correct the problem.  Repairs are made through an access panel behind the passengers seat.

Air Bag Warning Indicator:  Caused by a faulty drivers side seat belt tensioner.   Replacement of the tensioner and resetting the indicator is all that is required.

Paint Chips:  Probably the most talked about problem with the Z3 is paint chips.   Attributed partially to the new environmentally aligned paints and the aerodynamics of the hood that seems to attract debris just like the intake of a jet engine.  Chips have been found as far up as the windshield washer nozzles.

Additional information about reported problems can be found at the National Highway Traffic Safety site.