I'll take my Z just about anywhere and not feel too guilty about it.
I would probably say the most embarrassing for me personally though is pulling in for an oil change... Wish I could do it myself but it's against my lease with my apartment complex =/
However, I did drive it to work one day and while waiting for an older couple to walk across the parking lot in front of me I noticed them say to each other something along the lines of 'some people have too much money'. Being that I'm only 22 and paying for the car myself I couldn't help but laugh a little bit.