  We argue that weak containment is an appropriate notion of amenability for
inverse semigroups. Given an inverse semigroup $S$ and a homomorphism $\phi$ of
$S$ onto a group $G$, we show, under an assumption on $\ker(\phi)$, that $S$
has weak containment if and only if $G$ is amenable and $\ker(\phi)$ has weak
containment. Using Fell bundle amenability, we find a related result for
inverse semigroups with zero. We show that all graph inverse semigroups have
weak containment and that Nica's inverse semigroup $\mcT_{G,P}$ of a
quasi-lattice ordered group $(G,P)$ has weak containment if and only if $(G,P)$
is amenable.
