Usually: Worn sway bar links — by far the most common cause and one of the cheapest to fix.
But it can also be: Strut mounts, worn bushings, or something loose in the boot rolling around, which is worth ruling out first.
Clunking over bumps
A clunk that shows up over bumps, especially small ones, and is worse at low speed on rough roads, is most often worn sway bar links. They connect the sway bar to the suspension on each side, and once they wear out they develop a little play. That play is what you hear as a clunk. It is one of the more common noises we run into, and one of the cheaper ones to fix.
Before you assume that, though, do the easy check first. Open the trunk or cargo area and see if anything is loose back there — a jack, a toolbox, a case of water. It sounds obvious, but a rolling object in the back can produce a clunk that sounds exactly like it is coming from the suspension. Rule that out before anything else.
If the back is empty, there is a rough test you can do yourself in a driveway or parking lot. With the car parked and off, grab the top of each front tire and rock it in and out, and push down on each corner of the car and let it bounce once. A clunk or knock during that test points toward links or bushings rather than something loose in the cabin. It will not tell you exactly which part, since links, bushings, and strut mounts can all produce a similar sound, but it confirms the noise is coming from the suspension.
Worn bushings and worn strut mounts can cause the same kind of clunk, and telling them apart from sway bar links usually means putting the car on a lift and checking each joint by hand for play. That is not something you can do reliably from the driver's seat or in a driveway.
As for driving on it — a clunk from sway bar links is not a safety emergency in the short term. The sway bar's job is to control body roll in turns, not to hold the car together, so worn links do not typically cause a sudden failure. That said, if the clunk is accompanied by looser or vaguer steering, or if it gets noticeably worse over a week or two, that is worth getting looked at sooner rather than later.
Fixing it usually means replacing the sway bar links, sometimes on both sides even if only one side is making noise, since they tend to wear at a similar rate. It is a straightforward job, and the clunk is normally gone completely once it is done. If a look on the lift finds worn bushings or a bad strut mount instead, we will tell you that before doing anything, since the cost and the work involved are different.