What is the advantage of m·s^(-1) over m/s? It's everywhere, but no matter who I ask I can never get a straight answer, only "I don't know"s. What is the advantage of using multiplication and negative powers over just using division when writing out units, as in the question title. Granted, in most cases they use superscript rather than how I formatted it in the title, but my phone keyboard would not let me do that.

I don't have a problem reading units written this way, but it just seems inefficient as it uses more characters and makes it more difficult for "regular people" to follow. I'm suspecting it's nothing more than a style choice, possibly because it looks "more scientific".
I think the main advantage is that an ambiguity of a/bc is avoided. For example, g/m^2s may be read as g m^{-2} s^{-1} or g m^{-2} s depending on whether multiplication takes priority over division.  As far as I know, there is no consistency in opinions about it, so to avoid confusion a slightly uglier notation with negative powers may be worth accepting.