I'd finally gotten my wife's old commuter bike up and running when it started getting flats. I checked the inside of the tire, nothing. I checked the rim, well taped up, and nothing. No spokes poking through. Ugh. I still don't know what's causing it.
The tire is old, the rim tape is old. I'm going to replace them both tomorrow and plunk it all in with a new tube. By old I mean like 12 years old.
These are not pinch flats. Just pinholes, the most recent was on the rim side. Hence the new tape. Hopefully this will take care of it.