  Taking residual finiteness as a starting point, we consider three related
finiteness properties: weak subsemigroup separability, strong subsemigroup
separability and complete separability. We investigate whether each of these
properties is inherited by Sch\"utzenberger groups. The main result of this
paper states that for a finitely generated commutative semigroup $S$, these
three separability conditions coincide and are equivalent to every
$\mathcal{H}$-class of $S$ being finite. We also provide examples to show that
these properties in general differ for commutative semigroups and finitely
generated semigroups. For a semigroup with finitely many $\mathcal{H}$-classes,
we investigate whether it has one of these properties if and only if all its
Sch\"utzenberger groups have the property.
