The ice dispenser on my Kenmore coldspot model 10659702992 stopped working. Then the ice maker itself stopped working. The water dispenser works fine. What is most likely wrong?

Most likely, the dispenser's throat for the ice is clogged with ice or there is a large chunk of fused cubes blocking the exit.

Easiest thing is to pull the ice tray out and dump it, then check the throat to see if it is clogged.

If that doesn't work, check the service manual or call Sears repair.