We use real towels for all the "cleaner" messes and rags for all the other stuff. We have an elderly dog, a cat, and a toddler, so "other stuff" can be quite the category some days. But holey clothing can be torn up for re-use and then thrown away without regret, and without the need to purchase paper towels. As "real" towels age and get holes, they're downgraded to the rag bin.
We use the flat cloth diapers to clean up any and all pee messes. (Kiddo is mostly potty trained, but there are accidents.)