@Max...sorry to rain on your parade (or pee on your fire) but I would say 100% of all the brick type laptop and other power supplies I have ever taken apart (and there have been a few) have been switching types with the output isolated from the mains.
A mini-tower desktop computer might be different - those power supplies usually have a connection between the mains earth and the case of the PC....but I am not sure whether this will extend to the USB shield, I will check mine when I can.
Your solution might be to just switch your power supply off rather than unplugging it. Of course you should also check if the power supply ground goes all the way through to the output.
The best solution would be to get a mains cable with a proper 3-pin plug, cut the IEC end off (preferably while it is NOT plugged in :-), strip it back a bit, and cut off and insulate the white and black wires (which is what I think you have in the US) leaving the green earth wire only - and connect that to your anti-static mat. This does assume you have a spare power outlet where you are working, and that G the G does not want to use said outlet while you are electronicking...
PS to avoid the white and black wires shorting, cut them off different lengths. Then get some pliers and pull the insulation so you don't have wire ends sticking out. Then insulate them with tape. As the Irish would say....To be sure...to be sure...
@Martin So if your circuit is floating and you're connected to the low side of its power supply, you should be OK.
Also a good idea to connect a scope probe ground to a floating circuit as the first connection so that the circuit is no longer floating. I've zapped MOSFETs by connecting the probe tip to the gate first before the ground.
The purpose of the wrist band is to provide ahigh-resistance (leakage path) from you to whatever you're working on. So if your circuit is floating and you're connected to the low side of its power supply, you should be OK. If the power supplye is indeed connected to the ground lead of the AC mains, then that's better.
Some years ago, I was installing a new hard drive in my desktop PC. When I powered it up, the power sypply fan came on, but nothing else. I had zapped the motherboard and needed a new one. Since that day, I always wear an ESD wrist strap.