The Z06, and all C5s for that matter, do seem to have a few quirks to them.  They can be annoying at times, but when you think about the power and capabilities of the car (405hp from a 350 is damn impressive), you realize they are not an issue, and most race car drivers would feel right at home in our cars due to these noises and quirks.  Here are the most common ones, so that hopefully you can take comfort in the fact that the sound or characteristic you've noticed is pretty common and not a major issue.  You may even learn to love some of them :-)

DIC Messages:

Charge System Fault:  Once or twice when starting the car, no big deal, computer got out of synch with the starting cycle.  As long as the error goes away immediately and the battery hovers at 13v running and 12v not running you're fine.  Keep an eye on the battery voltage, as this can be a sign of the battery being on its way out or alternator/starter problems (corrosion).  If the message stays on or comes on while running, get it checked out.

Service Column Lock:  This can be serious, or a fluke.  If your battery dies, its probably a fluke.  Even if your battery didn't die, this will sometimes pop up.  The computer is very sensitive to the column lock's voltage and current draw due to issues with the mechanism in the past, so any abnormality will cause it to disable itself and throw the error.  If the column is stuck locked however, this is a more serious issue.  You can pull down from the top of the steering wheel (as if making a right hand turn) while turning the key on and off, or if that doesn't work jiggle the wheel back and forth, again while turning they key on and off.  That should help free it up, but after that go to the dealer, it is dying.  There have been several recalls.  If the wheel isn't locked, pull fuse number 25 in the passenger footwell and put it back in after 10 seconds, should reset the system.  Keep an eye on it though!

Other:

Exhaust jingle:  When decelerating in gear or downshifting you hear a jingling or rattling sound coming from under/behind the car.  This is actually a resonating whistling sound caused by air being sucked through the small hole in the factory H pipe (the link between the 2 exhausts) caused by increased backpressure when letting off the gas in gear.  Replacing this with an X pipe or even an H pipe with full size hole should eliminate the sound.  Or when coasting you can hold the clutch in, but that's no fun.

Brake squeak:  High performance (soft) brake pads cause this as well as increased brake dust in the front.  It is normal, and if its annoying, harder brake pads will solve both problems, but decrease brake performance somewhat.

Clunk from under car when shutting it off (often the car shakes too):  Just the transmission coming to a stop.  Happens when the car is shut off with the clutch not pushed in.  Try shutting off the car with the clutch pushed in, the sound should not be present.  This is normal.  Usually also happens to a lesser extent if the car is idling and you let the clutch out in neutral.  It just seems loud since you literally sit right next to the clutch and driveshaft.

Exhaust pops when shifting or decelerating:  Usually happens after a cold/cool start in the first 5-10 mins of driving, but happens sometimes even after running for a while.  Just extra fuel and carbon burning off, no worries.  However, if it happens a lot when the car is warmed up and you've been driving for a while it can mean you are running rich or one cylinder isn't firing.  A few times in the first few accelerations after startup or once or twice while driving is normal, but anything more consistent should be investigated.

Wipers "weep" onto the windshield at WOT or high speed:  No check valve and the laws of physics cause this, its normal.  If an issue a small check valve can be put into the hose (so it only lets high pressure fluid through).

Lifter noise (sewing machine sound under the hood, especially during warm up):  Normal due to higher lift camshaft.  Some also blame this on cold piston knock, there was a bulletin for this on LS1 but not LS6 I don't think.  The fuel injectors on our cars are also somewhat noisy and will make a "tick" of their own.

Recalls and fixable issues:

Wipers come on/spray randomly:  Wiper movement when powered off can mean the drain hole under the wiper motor is plugged up and is full of water, thus shorting the motor.  Wipers spraying can be a short in the steering wheel causing it to happen when you turn the wheel occasionally  If the wipers are stuck on, pull fuse #10 in the passenger footwell.  Regardless, any of these issues warrant replacement of the motor (unless you want to try cleaning it first), switch/stalk, and/or wiring in the steering shaft. 

Seat Rocking back and forth:  02 and 01 had a TSB (02-08-50-002) for this, there is a fix.  Some 03 have reported this issue also.  Just the seatback having give to it is normal, but if the whole seat (including bottom) moves, there is a fix.

01 oil consumption:  TSB (01-06-01-023A) issued to replace the rings and valley seal.  Early 02s may have the issue as well

Fuel gauge stops working:  Bad sender, has to be replaced

Clutch pedal sticks to floor or partway:  Seems more common on '01s, but even at that pretty rare.  If you get it, you have to fight with the dealer to get clutch swapped under warranty.

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