What is the (meter*second)/second, or ms/s unit used for?

I think you mean (meter/second) /second, or meter/second^2. It's a unit of acceleration, the rate of change of velocity. How much does speed (measured in m/s) change each second? That rate is measured in m/s per second, or (m/s)/s, or m/s^2.

m*s/s would just be meters. The seconds cancel out.

(meter*second)/second = meter, measure length

You ask what it is used for. The equivalent ( metres ) is a measure of length.
I do not know of any physical quantity that is represented by metres * seconds
Hence I cannot imagine any use for the unit as written.

This contrasts with the expression IMPULSE / time
the units of impulse come from Force * time
so it becomes (Force * time) / time
(Newton seconds ) / seconds
But the numerator DOES have a real meaning in this expression.