Well, it depends on what you're dealing with. I've repaired large tears in bed sheet with patches, or small holes (cat claws) just with embroidery floss.
For the large tears, I first put a press cloth over the ironing board, lay the sheet face down and line the edges back up as well as I can. Then I cover it with a strip of interfacing large enough to hold the edges in place and iron it down. That prevents everything from shifting. Then I flip it back over and cut a piece of fabric in a strip wide enough to give good coverage - round your corners. Pin it in place and do a wide tight zig zag around the edges. Ideally this will attach all 3 layers together (interfacing, sheet and patch). Then go back and trim any loose threads, any edges peeking out (both interfacing and patch), etc. I usually use flannel for this so it's nice and soft (make sure it's pre-shrunk).
With little holes, I just put the sheet in an embroidery hoop and darn it with a couple strands of matching thread.
