Out of nowhere, my favorite sheets developed three holes. (Perhaps it's all my nighttime tossing and turning?) I have a bit of a sheet fetish, so there was no way I was going to throw out my favorites. Also, sheets are expensive. I've done other basic mending before and this was pretty much the same process.
At first I tried just sewing (using a whip stitch? I'm not great with stitch names) the two sides of the tear together, but the fabric was too worn, so I cut out my stitches and used some iron on hem tape to reinforce the back of the tears with another little scrap of fabric. Then I re-sewed the two of the holes with the "whip stitch".
I didn't really love how it turned out because the fabric was a little puckered and it looked like the stitches could pull out.
With the final tear, I reinforced the back of the sheet with scrap fabric and the hem tape, but then used a satin stitch to cover the tear. I was much happier with how this turned out. It didn't create any wrinkles in the fabric and it seems like a much stronger repair.