I have some bath towels that have frayed on the ends and I decided to restore them and use them for assorted other tasks.
You can set up the serger for a 3 thread picot edge. (Rolled hem settings with a longer stitch length.) Cut the towels to wash cloth or tea towel size and serge the cut edges. They can now be used for dusting, wiping up spills or anything you want. If you like you can leave a long thread tail on one corner and catch the end of the tail in the new serged side and use it as a hanging loop.
When old sheets have gotten tears or frayed I cut them up and make pillow cases.
Of course assuming that the remaining sections of the towel and sheet still have some miles left.
Hope you are inspired to recycle and reuse....