I finally found what is blowing up my MOSFETs: the PicoScope.
The PicoScope's power supply is placing 30 Vac onto the PicoScope's ground (!), which is connected to the circuit I am developing. Whenever I solder a MOSFET in place (with a grounded soldering iron), that's putting 30 V across the MOSFET, blowing it.
So: if you're using a PicoScope, make sure to ground your circuits directly to earth ground before you solder them!