Yes and no. Even singlemode can be pretty flexible. Every manufacturer's product is different, but even 20 years ago, the singlemode we were installing could be wrapped around a Sharpie without damage (though it would lose a lot of light; that was an installation-only ability), and would happily tolerate something like twice that bend radius during operation.

That being said, I've seen people do some horrific things with copper, and it tends to survive things it shouldn't. But if you treat glass even half as gently as copper wants to be treated, they'll both be fine.

I think fiber gets a bad rep because large multi-strand cables have stiffeners to keep the jacket from kinking, and people assume the individual strands are similarly stiff. They really aren't.