How is Study Abroad regarded by PhD admissions committees? Does study abroad help enhance a PhD candidate's profile or does it not carry much weight at all?  Let's say I took "regular" coursework while on study abroad for a semester, like a core course and not something research-focused.

Would my having lived in Prague or Florence or Shanghai be something that PhD admissions committees consider favorably?

The field is applied mathematics, statistics, or operations research.
I think this is too broad a question to have a really good answer, but I think the most generic one is: not really.  If your study abroad experience could provide some unique opportunity to deepen your knowledge of the field, then that might get some real consideration.  Otherwise, it probably won't be seen that differently from other classes, and in some cases viewed more skeptically if they were at a program whose reputation is less established.  EDIT: In particular, to address the last sentence, living abroad will count for extremely little.

As one counterexample, I will mention that when I applied for grad school, one graduate director explicitly mentioned my study abroad experience.  However, I think this is the sort of exception that proves the rule: I went to a very well established program specific to my field; I was going to a small liberal arts institution, so the study abroad program actually had a stronger reputation, probably; and I took a very ambitious schedule and he mentioned the specific classes I had taken, not the generic fact that I had gone.

My point is, study abroad can mean many things.  Use your good sense (or talk to mentors) about how different ones can affect your graduate school applications.