It depends. I'm not a fan of it simply for "the look". I think it looks kind of messy in its own right. But I always like it when people personalize their style, which may include a favorite scarf, strange keychains, etc added to their bag.
It depends on the bag, the scarf, and the person carrying it. I think it can look very chic on the right woman and go terribly wrong on the next.
ITA. It's *so* not a look for me, though I love seeing it. I think it has to be done with confidence and nonchalance to not look like you're trying too hard.
__________________
"Good taste shouldn't have to cost anything extra." - Mickey Drexler
I think its adorable when you have a plain bag and just want to accessorize it a tad with the right sized scarf. For some reason, I feel like its more of a spring/summer look though.
For me this is often a practical choice. I may start out with a scarf around my neck or jacket lapels, but then it bugs me or the weather suddenly turns warm, and I'll loop it around my purse because it feels safer than stuffing it inside.
Looking at Zeitgeist's examples makes me think that a scarf with a slight bit of airy volume gives the best look- -especially if you're going more classic/traditional. I like how you color-matched with your bags.