I much prefer rolled. It lasts longer.
Handles are one of the biggest stress points on a bag; and flat ones tend to split at the "fold".
Hasn't happened on any of mine yet, but I clearly remember my mom's and grandmom's.
Chain. Only chains, entwined with a little strip of bag material. This is currently the only bag suspension system that I find acceptable. This phase has already lasted several months. Who knows where it will end?
I make exceptions only for the one-of-a-kind hand-embroidered pieces of wearable art which usually have cloth straps, though one of my most beautiful, and I guess in all fairness, most valuable, has a strap made of something like nylon webbing, but I never gave it any thought, I had to go look at them to see!
I prefer flat for a shoulder bag. Most rolled handles I've encountered are too stiff and won't stay up on my shoulder. But, of course, rolled handles are better and more durable for an arm or hand-held bag.
I think it depends on the size of the bag to which the bag is attached. I have rolled straps that are very comfortble, as well as the flat braided straps. Don't exactly know how to respond because each bag is different.