Today when I was turning in a parking lot I heard my tiers chirp. It made the sound whether I was accelerating, braking or coasting. It only did it when I was turning to the left. I repeated the turn several times to figure out where it was coming from but I couldn't tell for sure. I think it was coming from one of the back tires but it seemed to switch from side to side. I got out of the car to check under all four wheels but nothing seemed amiss. When I turned to the right, I hear a clunk and then everything was fine. The chirping went away and the car was behaving normal again. It never came back.
I suspect this had something to do with the differential. Maybe it was staying locked when it should have let go. I’ve never changed oil in the differential and I suspect the previous owner hasn’t either. Would that fix it? Has anybody else had this problem?
Thank you all for your help.

94 965 80K

Sometimes the front tires will make noise when you turn full lock or nearly so and the stripes in the lot are freshly painted and smooth.

I drive a '96 Chevrolet police cruiser that turns its wheels so sharply that one of the front wheels slides almost as much as it rolls. On fresh striping, the sound is a squeal if along a line or a chirp if across a line.
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3 8s & 2 7s 740,000 miles total

Personally I'd have the rear suspension looked at for a bad bushing or a broken or bad piece causing alignment problems.

The differential is not staying locked; if that was happening, it would be a problem that you wouldn't be able to ignore. In fact, you'd have a hard time driving around turns.
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Chris Herbst, near Chicago.

Thanks for the reply.
I will have the rear suspension checked out for any problems just to be sure but I’m not convinced that's the source of the problem. The suspension is working perfectly fine otherwise. There are no noises or vibrations, it tracks straight at any speed and soaks up all the bumps. Also the rear suspension on my car is a live axle so the wheels are always aligned relative to each other no matter what. This has only happened once, and it went away as soon as I turned the other direction. I couldn't replicate the condition again.

94 965 80K

The live axle isn't necessarily aligned relative to the front wheels. It is when everything is good. if something broke, it doesn't necessarily line up.
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Chris Herbst, near Chicago.