  Given any quasi-countable, in particular any countable inverse semigroup $S$,
we introduce a way to equip $S$ with a proper and right subinvariant extended
metric. This generalizes the notion of proper, right invariant metrics for
discrete countable groups. Such a metric is shown to be unique up to bijective
coarse equivalence of the semigroup, and hence depends essentially only on $S$.
This allows us to unambiguously define the uniform Roe algebra of $S$, which we
prove can be realized as a canonical crossed product of $\ell^\infty(S)$ and
$S$. We relate these metrics to the analogous metrics on Hausdorff \'{e}tale
groupoids.
  Using this setting, we study those inverse semigroups with asymptotic
dimension $0$. Generalizing results known for groups, we show that these are
precisely the locally finite inverse semigroups, and are further characterized
by having strongly quasi-diagonal uniform Roe algebras. We show that, unlike in
the group case, having a finite uniform Roe algebra is strictly weaker and is
characterized by $S$ being locally $\mathcal L$-finite, and equivalently sparse
as a metric space.
