Keep a Fitted Sheet from Slipping with Velcro
First, you'll need some industrial strength velcro. It helps to sew it in, but the adhesive kind can work if you stick it to a flat edge of your mattress. You want to put the rough part of the velcro on the mattress and the soft part on the sheet. You'll need to at least sew on the sheet portion of the velcro will come off in the wash. Put velcro on all four corners but pick one corner to have a unique shape or something different from the other three. This way you can tell which corner matches up to a certain corner on the bed. For example, you could make a dot with a permanent marker on the left-bottom corner's velcro.
This might be a bit of work up front, but shouldn't take you more than 20 minutes to accomplish. When you're done, you'll have a fitted sheet that stays put and can tell you exactly how to put it on your bed.