Any suggestions on the easiest way to sew elastic on a bed sheet? Do I need to stretch the elastic and hold it in place while I sew? If so, how do I avoid interfering with the action of the feed dogs? Many thanks.

 

What sort of elastic are you using? I am gathering (sorry about the pun) that you are repairing an existing fitted sheet? Recently I repaired a fitted sheet using fitted sheet elastic which is designed to fold over the edge. I just stretched the elastic to almost full extent and stitched through all layers.
I also repaired a sheet without using the fitted sheet elastic. I cut a hole in the existing sheet elastic and threaded a thin elastic into the casing made by the original elastic. I only started just before a corner and ran the elastic up the side to past the next corner. It was repeated on the other side.

Even just the corners can be done.

I've never worried about the feeddogs - Only when I sew wadding do I put a piece of fabric, tear away or even paper between the wadding and the feed dogs.