For a Banach algebra \(\mathcal{A}\) which is also an \(\mathfrak{A}\)-bimodule, we study relations between module amenability of closed subalgebras of \(\mathcal{A}''\), which contains \(\mathcal{A}\), and module Arens regularity of \(\mathcal{A}\) and the role of the module topological centre in module amenability of \(\mathcal{A}''\). Then we apply these results to \(\mathcal{A}=l^{1}(S)\) and \(\mathfrak{A}=l^{1}(E)\) for an inverse semigroup S with subsemigroup E of idempotents. We also show that l 1(S) is module amenable (equivalently, S is amenable) if and only if an appropriate group homomorphic image of S, the discrete group \(\frac{S}{\approx}\), is amenable. Moreover, we define super module amenability and show that l 1(S) is super module amenable if and only if \(\frac{S}{\approx}\) is finite.