All good messenger bags come with two straps. One over the shoulder to carry the weight and another that goes around your waist to stop the bag from slipping around to the front. If you are using both straps the bag shouldn't come off your shoulder when riding.

I have used a largish fanny pack when riding to work a long time ago. It worked okay, but I prefer to wear a backpack as I don't enjoy having my waist constricted with the the main strap on a fanny pack. Depending on your size requirements there are quite a few packs these days that stay off your back to keep it fairly dry - see pic below.