What is the advantage of m·s^(-1) over m/s?
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.
Generally speaking, the choice of one form over another is better done on a case-by-case basis, according to readability.
If I have to specify a speed of 5 meters per second, I'll write v = 5 m/s rather than v = 5 m·s-1, because for most people the first form is more readable than the latter.
However, if I have to label the axis of a quantity with some complex unit, I may choose the form with the exponents. For, instance, if I have to report the spectral density function Sv(f) of some voltage noise, whose unit is V2/Hz, I may label the vertical axis (according to the style I described here) either
Sv(f)/(V2/Hz) (with parentheses to avoid any ambiguity)
or
Sv(f)/V2Hz-1 (for compactness).
In this case, I'd probably choose the second form because is probably more readable even in the case of more complex units. Strictly speaking, also this second form without parentheses is ambiguous because multiplication and division are associative from left to right, but frequently people understand that unit as a single block.
Remark. If you use LaTeX to write papers I cannot but recommend to use the package siunitx to typeset the units, because it allows you to switch between the two forms with just one option (per-mode=symbol or per-mode=reciprocal). Moreover, it automatically adds a small space between the units.