Rear shock mounts can take a beating on the road, really they are a wear item that you should expect to have to replace at some point. Specifically, let's take a look at the 8-Series E31 setup which is similar in respects to the E30 and E36 3-series with the shock mounted separately from the spring. This is a key difference from the later designs such as the E38/E39 where the rear shock sits inside the spring.
But first, how do you know you have to change the mount? If you can see visual damage then it’s a definite - jack up a corner and just have a look in the wheel well. If you can see the inner rubber being pulled down or detached in any way it's a sure sign of failure. Failing mounts also tend to be responsible for squeaking noises and increased vibration in the rear.
The assembly can all lift out as one unit

The good news is that changing out mounts on these cars is an easy DIY job. Securely jack the car and remove the wheel. Undo the large bolt at the bottom of the shock absorber and the lower arm will drop down a little further (it’s still held securely by the upper arm and trailing arm so don’t worry about this). You can now go into the trunk and undo the two nuts holding the mount at the top. The shock absorber, bump stop, and mount can all now lift out as one unit.
Then just remove the top nut on the shock (usually by holding the inner core with a hex socket to stop it spinning), remove the mount, replace with new, and reassemble.
I specifically called out the E31/E38 difference above because on the E31 and related the lower arm can be dropped with the spring staying in place. On the E38/E39 a spring compressor must be used to remove the spring before the shock absorber can be freed. Remember - compressed springs are dangerous! Changing the mounts on those cars can be a lot trickier for this reason so don’t try a DIY unless you’re confident with the spring removal.

Reinforce the chassis
Whilst you have the shocker out of the car examine the metal around the mount point for any signs of fatigue. The worst failure case is when the mounting area actually snaps from the chassis leaving the shock absorber top floating in free air! Reinforcement plates like in this E36 Shock Mount Kit are available to prevent this catastrophe from occurring.