Fusible applique is probably the most likely to fray from many washings, so I'd avoid that. The exception would be if you do a lot of machine quilting that crosses over the edges. If you like a frayed look, you can do raw edge applique without a fusible and just machine stitch near the edges.
Most likely what you want, though, is a turned-under-edge form of applique.
Probably the most stable form of turned-under applique is machine straight stitching on the edges of the applique (one-sixteenth of an inch or less from the edge); however, very few people do this.
Hand applique would be fine. You would want to lock stitches periodically (knot or double-stitch in place) so that if a thread breaks, only an inch or two of applique edge needs to be repaired.
I like to use machine applique a la Harriet Hargrave or Sharon Schamber techniques. Invisible thread holds just as well as cotton thread for this technique, I think. Just stitch in place a few times at beginning and end to secure. If you are a perfectionist, you can pull threads to the back and knot before cutting. I wouldn't do this for a quilt that's going to get used a lot; it would be less time-consuming to simply hand mend any spot that might come loose later (unlikely, but it could happen).
My two cents, anyway.....
Mary