I have among many cars, a 2005 Chevy cobalt LS. it's a nightmare! Spend more time under it than driving it. So recently I replaced the front passenger ball joint, and the driver's side rear coil spring. The area I live in is pothole h311. Suspension and steering problems are constant. Now I have a loud clicking/thumping sound coming from the rear end. I'm not positive but I think it's the drivers side. It usually makes this noise when first taking off after several(or more) hours of not moving. After about 5 minutes it seems to stop, however it will start again randomly or sometimes when turning. It's very unpredictable, it's getting progressively louder and recently has started shaking the car when making said noise. Sometimes it's the clicking sound similar to a bad CV joint, others its more of a thumping sound like a flat tire. The alignment is dead on center, the steering wheel doesn't pull ever, no shaking in the or vibration either. I want to say it's just a broken belt in the tire, however the fact it isn't a constant continuous problem makes me think otherwise. Any thoughts? Possibly a bad strut??
If your tire had a bad belt or a problem that could shake the car it would be 1) visible and 2) constant. As this is a front wheel drive car there's no rear axle, diff or CV joints, so the problem is almost certainly in the rear suspension, which is a torsion beam with a spring and shock. You've recently disassembled the rear drivers side to replace the spring and that's where you think the sound/vibration is coming from, I'm going to hazard a guess that the problem lies there.
The very first thing I'd check is that the wheel has been bolted on properly, it's one of those id10t errors that anyone can do. Check the other side as well, it will take a few extra seconds.
If the wheel is on right then you need to check the suspension. It's possible that when you replaced the spring you didn't re-assemble the suspension correctly. It's also possible that some of the bushings have degraded and couldn't handle being re-used, or that another component has failed since. I'd jack it up and waggle tire around, see if there's play. Get a pry-bar or breaker bar in there and find out what's broken or come loose.