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AC54ME's summary is spot on. Some of the 01s used a lot of oil although the TSB is for the Z06 so a Vert shouldn't be affected. Even so, an 02 may be a better option if you can find one.
The only other thing I've had on my 02 is a bad door lock module. I'd say battery and diff are the ones to check carefully.
I just purchased a 2000 Vert and really love the ride. But, it does have problems as noted above. The tranny is prone to leak after 50k or so. I guess I have been lucky for in all my 55 years since I have never owned a car that had a tranny or rear end leak? Oil leaks, yes. But never those two. Now, I have them both and the dealerships seem unable to repair them.
So, watch out for those two areas for they are expensive to fix. I also have experienced a code or two suggesting my ABS system was having trouble. This one can be expensive to fix too.
I've got a 2003 with 25K miles. I'm getting all kinds of messages about the ABS system, Traction Control System, etc. It's a C1214 error code. Took it to the dealer and they want 2800 to fix it saying the EBCM and BPM are both bad. I checked this forum and evidently this may be a common problem on C5 vettes.
I'm not paying 2800.00 to fix it, instead I've removed the EBCM and hoping the guys at ABSFixer.com can repair it. If not, I'm out a lot of money.
My situation isn't unique if you do a search for EBCM on this forum.
Good luck.
I had the same messages appear when my Vette was the same age and milage as yours is now. ABS TC AH etc would light up. I could turn the car off and wait a few minutes and turn it back on and they went away. ended up replacing the battery and have never had a problem since. My Vette is nearly 8 years old and 60,000+ miles. No other major problems to report.