1996 Nissan Sentra GXE, 147,000 miles. Automatic transmission.
When I start moving after starting the car, I hear an ominously loud single clunk coming from the area around the right rear wheel.
I start the engine, no clunk. I release the parking brake, no clunk. I put the car into gear, no clunk. I take my foot off the brake and begin to move, CLUNK!
And that’s the last I hear of it while I’m driving. There are no rattles, shakes, or other problems as I am driving. I can stop for a traffic light, and there’s no clunk when I start moving again. Shift into reverse, and there’s no clunk. Hit a pothole, nothing.
It may be worth mentioning that I have a huge hole in the right side of my muffler (seriously - I can stick a few fingers into it) which I am not going to repair.
And that is because in five weeks I am going to be buying a new car. I’ve arranged time off from work to do all the running around from bank to dealer to insurance agent to DMV, and I’ll have enough cash at the ready for a large down payment.
So I am very reluctant to have any work done on this car.
So what it comes down to is can I ‘baby’ the Sentra as much as possible and use it as little as possible over the next few weeks, or am I in danger of losing a wheel (or something) at any moment?
There’s no way to tell without getting under the car. It could be as insignificant as a worn out sway bar bushing or as serious as a ball joint ready to come apart. Personally, I wouldn’t drive it until you find out what’s causing the clunk. If you live in an area where public transportation is available, use that until you get your new wheels. If not, perhaps a bicycle of the generosity of friends.
Does it do it every time you start to move or only after sitting x amount of hours? Disc or drum brakes in the rear?
Does this old Sentra have ABS?
I ask because, based on your description, the noise only seems to appear at the point when an ABS system does its self-test (first forward movement after a cold start).
How long have you owned this car?
Does it have ABS?
Do you use the parking brake a lot? You park on a hill every day or something like that?
Have you tried parking it without using the parking brake overnight?
This sounds to me like the rear shoes are sticking, probably a weak or broken spring.
If this is the problem, you should be able to baby it for a while. If you were going to keep it, I’d recommend fixing it. However, if the sound changes at all (turns to grinding, crunching, whatever when you actually use the brakes) then you’ll need to get it taken care of, as it will have just turned into a major safety issue…not being able to stop, or having the wheel lock up on you.
As mountainbike said, use it as little as possible, and start riding with friends. 
Hope this is right, and that it helps.