The notation m s^-2 and kg m^-3 always strike me as perverse: what was
wrong with m/s^2 and kg/m^3? I remember being taught m/s^2, and then
when I changed school to one that did a Nuffield physics syllabus, the
notation changed to kg m^-2 which it claimed to be "better" in some
unspecified way.

its to do with the fact that mathematically speaking:

1 / x is equivalent to x^-1

So therefore

1/x^2 is equivalent to x^-2.

so once the 1 is replaced by another quantity we have:

for density say:

kg / m^3 is equivalent to kgm^-3