Symptoms include no audio inputs when a headset is plugged in, but the built-in mic works fine when the headset is not plugged in.
sudo killall coreaudiod is one suggested fix.
This superuser.com article tells you how to reload Mac audio drivers without rebooting.
Thanks to Mohan Kartha for helping sort this out; still not convinced I can adequately fix it with these tools, or even that I have diagnosed the issue. Some of the discussion is about a "stuck optical switch".
Genius bar next.
UPDATE: Genius bar helpfully scheduled an appointment. The folks at the Apple store helpfully let me try other headsets and other mics and other computers, and the problem seemed to be isolated to my computer.
In the meantime I poked and poked and eventually landed on a description of the "Audio MIDI Setup" utility. I dutifully clicked through to it to see what was up, adjusted a few things, and then found this:
Note that audio input is unhelpfully marked "mute". When I unmute, everything works fine. Why on earth this only shows up deep in the MIDI setup of all places I'm not sure; I know that I had fiddled around there before.