Lighting – Light bulb spins in recessed but won’t unscrew

Any number of things.

1. Most recessed light fixtures contain what is called a thermal cutout. These cut off the power to the light when the heat from the light exceeds the temperature rating of the fixture. They go bad.
2. Sockets go bad
3. Switches go bad
4. Loose wire connections inside the fixture.
5. Loose wire connections behind the switch.
6. Loose wire connections feeding the switch from another source.
7. It is not uncommon for a wire staple to penetrate the wire and over time the wire burns up inside the wall.

If the last light goes out when the second to last is removed, the wiring error may be:

Where electricity travels through the second to last bulb, on it's way to the last bulb. If so the last two bulbs may also have been dimmer than normal (if they are incandescent).

The bulb type matters here: using CFL or LED lights during testing can add to the confusion.