My brother has an 87 Ford Mustang that he said is starting to make clunking noises when he turns or goes over bumps. He knows even less than I do about cars. I'm thinking shocks or tie rods? What could it be?

When he turns or goes over bumps? I think you have it right. Toss in suspension joints.
Bad tie rods are particulary dangerous. If they part you loose all control of the car.

I'm not sure if 87 Mustangs had sway bars, but my aftermarket one was loose and made a clunking noise.

I don't have a Mustang, but I've had to tighten the sway bars in my car twice now, and the first sign is the noise I hear when turning (a clang/clunking) while going slow and/or going over speed bumps, etc.

I don't know jack about cars, but I've heard that a clicking noise going around turns can be the CV joints (not that I know what a CV joint is).

I've heard of those before but like you, don't really know what they are. I told him to take it to a garage and have them check it out. He thought that by listening to it or driving it I would know what is wrong. I'll let everyone know what it turns out to be.

Likely suspension, if he hears it during while not accelerating or shifting gears. I don't know much about that generation of mustangs but AFAIK an 87 is a RWD car (no CV joints?). It could be a worn u joint, but you'll typically hear that under acceleration or when the car shifts into gear. Could be worn bushings on a sway bar or ball joints/control arms. If you'd like I'll ask in Mustang forum. Does he hear it in the front? on one side more than the other?

Well, according to him, the garage said it was the stabilizer on the driver's side and both got replaced and something that connects the front end to the wheels but he couldn't remember the name (tie rods?). So it's fixed and running fine.