Goat head thorns?
It's been my experience that those mofos can get through "puncture resistant"
Continental Contact tires, even with Mr. Tuffy liners installed. I've had it happen that I'll run over them, puncture the tube, then they fall/get knocked off the tire before I realize they're there. I hate those g'dmn things.
Of course, that wouldn't explain the rim-side punctures.