I just ran into this problem and thought I should write something up on it. I am finishing up a purse made from 1-2 pairs of old jeans. Because the denim is quite thick, in making tabs to hold the strap and purse together I could not fold the tab under to prevent ravelling. So I just zig zagged the tabs to the bag. Hmmm. I didn't like that look so - I looked in my ribbon stash and found some appropriate fabric ribbon. I removed the wires (don't forget some of these are already serged to prevent ravelling) and used the ribbon over the zig zag stitching. I could use this because the ribbon was thin and the material was flattened somewhat with the zig-zag stitch.
Don't forget to check out ribbon when it's on sale.