So on the way home from the parts store (with new bulbs for side and front markers, and new wipers), the TPMS light comes on. Checked all four the next day, and the left rear was down to 14PSI. Rolled the car back and forth but saw no nail/screw or tread damage. The following day. dead flat.
Luckily the factory jack and tools are in the trunk of our just-acquired 174k-mile 2004 coupe, so I jack & remove the wheel, place jack stand, and head to Firestone (closest place) just before business close. The desk manager believes it's a loose bead due to seasonal temperature change, and has me sign a work order for $20 flat repair.
I call the next morning at 9:30AM (having heard nothing) and the current desk jockey has decided that they don't have a machine capable of handling 35-series tires, and that I'd be better off at a generic rim shop or Discount tire (funny they couldn't tell me that the night before). I have my son call Discount tire, confirm they'll look at it, and run the wheel over there. When I pick it up after work, the manager there tells me it was a TPMS stem leak; they installed a TMPS stem repair kit & re-balanced, and I'm good to go AT NO CHARGE.

He may have been eyeing the 40% tread and looking forward to my business on a $1000 new set, but I was sure grateful for the correct troubleshooting and no hit to the teenage wallet. And my son got to rehearse his tire installation skills/get familiar with the supplied jack & iron on the flipside.

Wife's 2009 Wrangler Unlimited has had TPMS issues too - the aluminum stem on the spare's TPMS corroded away to where it went flat and would not take air (Firestone replaced for $100 with a rubber-stem one); other metal stems have cracked hex nuts, and I had Sam's club replace a TMPS that was broken off via a teenage accident (hence my hurry to get the spare fixed) for a low $55. But the system has still not recognized that rubber-stemmed sensor (it registered the Firestone-supplied spare sensor within a ten miles) so back to Sam's I go...

The TPMS system is helpful and expensive/aggravating in roughly equal measure, methinks. Curious what tire shop service experiences others have had with these specialty tires/TPMS.