Wobbly Rear Wheel
 

My back wheel was noticeably wobble on my ride home today where the whole wheel would shift itself. Just for clarification, the rim is not bent and I have the side bolts completely tightened. An easy way to describe it is that it looks as if it's loose on the axel that it rests on.

If its a sudden development, check for broken spoke.

The axle could be broken and need replacing . Once you remove the wheel from the bike you will know for sure .

Check the hub for play. Could be loose or it could be broken.

Sounds like the cup and cone bearings are worn or in need of adjustment. If it's been many years or not at all since you stripped down the wheel and cleaned and greased the axle bearings then it sounds like it's high time to do so now.

As part of the exercise of taking it apart, cleaning and checking things over prior to re-greasing and re-assembly you'll be able to check on the broken axle scenario.

Sounds like the hub to me. Might be able to adjust it, but I had a hub break and grind down recently. $ 42 later I had an entire new rear wheel. Weinmann 519 for a 26" atb. Wasn't happy that the old one broke, but relieved it didn't cost more and it was a 5-10 minute swap out and the bike rides as new again. The LBS tried to match up the bearings, but the inside bearing cup was jagged on the gear cluster side. I made that decision to replace it all and don't regret it. Not sure if they could've fixed it, but didn't want to waste the guys shop labor time to get something that may not hold up for the ride home ?