My car is a 2005 C6 with LS2 engine.  Problems are as follows.

1.  The most vexing problem is the "Service Active Handling System" fault and DIC notice.  It occurs randomly and sometimes locks out the cruise control when driving and locks up the shift when shifting to Park.  Then I have to start the car using accessory mode.  The dealer had tried 3 times to fix it but cannot find the cause.  So, I live with it.  Other than above, it seems to have no affect on driveability.   Sometimes, it goes away when restarting the car.

2.  Frequently, the shift lock locks the car in "Park".   When that happens, I have to use accessory mode and start the car in Neutral, then shift to D or R.   Many owners have this problem and have resorted to cutting the shift lock pawl.  I have not done that because I don't feel comfortable removing the console cover when I can simple use accessory mode to bypass the problem. 

3.   Recently, at 47,000 miles, I had to have the differential seal replaced on the drivers side axel.  Minor problem.   I also replaced the rear sway bar links.  They were very loose and one was dry and scratchy.  Haven't done the front yet.

4.  Tire wear is significant.  When alignment is set to "street use" specs,  I get severe wear on the inside edge of the front tires.   I got 21,500 miles on my last set.  I am attempting to get that corrected before buying another set of tires.

5.  Replaced all break pads at about 43,000 miles.

05' C6 Coupe, LS2; I always use Chevron gas with techron, and the last 4 times I filled it up, it would cut off with only 8 gallons or less, and acted like it was full, but it would only show 1/2 full on the gauge.  It was consistent, and I couldn't force another ounce into the tank.  A corvette shop recommended I bring it in and he would drop the rear axel and the right side tank to change out a crossover pump that feeds the separated tanks, and that was the problem.  I read on a corvette forum of people using "Sea Foam" in the system to correct this problem.  So, before I spent the money to overhaul the fuel system, I tried the Sea Foam.  I added it when I filled up, and again it only let me have a half tank, but when I ran the 16oz can thru the system, ran the tank down low, and refilled it, it filled all the way, and the gauge read full.  It's worked out for several months now, and haven't had the problem since, and I'm using the same gas station and same pump.

2009 Coupe. Manual Trans. At 8k miles, shifter would hang up between 1st and 2nd. Had to jiggle shifter and pull hard to get it into 2nd. Brought it to dealer, fixed under warranty. Said it needed 'adjusting'.

Body Control Module replacement.  There is also a sensor for the traction control system and transmission that may need to be replaced.  Our 2007 Coupe began exhibiting these same symptoms and was finally corrected with the total replacement of the BCM...$450-500 repair.

In regards to the fuel door not closing,, I found a spring inside the left rear interior panel where the cable release is for the fuel door. Replaced the spring with heavier one and works perfect... to try it at first I used a rubber band... 

Are you aware of CAGS (Computer Aided Gear Selection)? To help GM meet fleet fuel economy standards, CAGS locks out gears 2 and 3 during casual acceleration and forces shifting from first directly to fourth gear. Did the service people mention if your problem was related to CAGS?
 You can buy an aftermarket kit which can bypass CAGS so that you may shift to 2nd and 3rd under any rates of acceleration. Its installed under the car at a wiring connector.
Incidentally, in the C7 which has a 7-speed manual transmission, CAGS will force shifting from 2nd to 5th under casual acceleration.

There is also a circuit breaker change that you can do to block out the 1st to 4th system.  I got mine online but I'm sorry I have forgotten the name of the supplier.  You may be able to google it.  I have a 2007 Z06 and it worked fine on the car.    I had the same thing done to my 2002 Z06 by a dealer in Portland.  I think they did some type of electrical bypass to that one.

Both sun visor mirror covers just popped off. The plastic hinge pins cracked on their own. $500 to replace the visors. Cheap part for a $70k car. Found a $40 aftermarket cover that works ok.

In June of 2013, after 3 months of ownership, I had a Low Oil Pressure message on the info center also will zero oil pressure gauge indication.  I did some on line work to find out that there is a GM tech directive on all LS engines 2007 and later to have the oil pressure sensor fail.  The oil pan needs to be dropped and replace the sensor.  Work covered under warranty.  Job would normally be $1,300 out of pocket.

Wasn't the 2nd gear skip. Had trouble going into 2nd even when I had my foot in it.