I think that it helps to starch the fabric first, which also acts as a bit of a stain gaurd too
It is usually best to match the stabilizer with the fabric. Cutaway is the stongest. I use a product called cotton sheeting, which has no fray and is thick. Also peltex is very stiff or regular stiff interfacing. Lots of choices, the best thing to do is test, test, test/
I like a crisp , med to heavy wt cotton and a med wt cut away stabilizer. If fabric was flimsey I have used a heavy wt stabilizer./Lillian