I shut off all the breakers except for the bad circuit. Then I followed that cable in the attic with my non-contact volt tester, until it disappeared down a wall far away from where the power was out. I was about to give up, when later that day the lights came on and the outlets began working again (for a few minutes), even though the circuit I thought was bad had the breaker turned off (and the other breakers were turned on).
I was following the wrong cable! The outage was part of a larger circuit that contains 7 outlets and 3 light switches. The "red herring" circuit contains nothing. It is just power leading to behind a wall to no devices that I could find.
Once I determined the correct circuit, it was still tricky finding the bad outlet, because it had power, and there were no loose wires or signs of burning. It just didn't deliver any power to the other devices in the chain. Some internal defect in the receptacle.
The real villain in all of this is the painter, who painted over the breaker box and covered the labels. He also painted over all the outlets, which should have been replaced long before I moved in.
[This is a rental unit, but I have been doing all the maintenance, because the repair men they send out are unqualified and unprofessional.]
The bad outlet is the one used for the wall A/C, which is rated at 1050 watts. My guess is that the A/C over time had begun to burn something out in that receptacle, and the space heater finished the job.