DO NOT wash shop rags in your wife's washer. DO NOT.
Get a large plastic garbage bag, grab a baggie of powder detergent (or bottle of liquid) and head to the laundry mat. Late in the evening, if possible.
Find a corner machine a long ways away from any women in the place and discretely place said rags in the machine, add about twice as much soap as you're supposed to, and wash those beotches in the hottest water they have. Twice, if you're so inclined.
Then....get the plastic garbage bag and nonchalantly place the rags back in the thing and slip quietly out the back door, before someone smells the machine. Dry the things on a drying rack, or some facsimile of, at your house.
Yes, they'll have to run a couple of cycles in the thing to get it clean, but they know full well how to do this. Some laundry mats actually have "rag" machines that are dedicated to just this use!
Whatever you do, don't go in there in broad daylight and start throwing those filthy rags in a machine next to some laundry feminazi. It won't be pretty if you do.