If the lube or solvent is anything petroleum based, it can't go in compost or recycling, so garbage, this is for paper towels that I use. If a lot of solvent on them, I will spread them out on concrete in the sun with a rock or wrench on them and let them dry before putting in garbage. If cloth towels or rags, same thing for me, as the oil would just dirty my washer inner drum and/or clothes, this is not industrial cleaning. If I had access to "shop rags" like when I used the metal shop at my old employer, those go into the fire-safe (tight-lidded) shop-rag-can, where they are periodically sent out to get washed at the supplier for those, and returned clean.