I would check out the rear body mounts for age, mine were pretty much rusted and cracked, else just wear on exhaust for leaks and if its a ttop check out the rubber, you not get rained if only washing but doing a little speed, its always odd how a drip gets in.
Brakes and rear end problems, specificaly the wheel bearings back there. Funky noise, grinding in the rear, funny tire wear, tires tilted at odd angles, lots of play, noisy and/or weak brakes.
I could go on for hours...
Look especially well above the 2 gussets that are welded to the bottom of the frame in front of the rear wheels. The gussets form a sort of crap collector above there and that is where you will most likely find rustout. Anything mechanical can be fixed easily, but to do a frame off repair you have to go all the way. Paid a grand for mine, probably will have 10 in it before it leaves the driveway, not including a paint job.
"Chevrolet Corvette Restoration Guide" by Lindsay Potter, MBI Publishing Company, has a very good chapter called, appropriately, "Buying a Used Corvette". It contains a lot of good advice on what to look for, what to avoid, etc.