Let $G$ be a locally compact group, and let $L^1 (G)$ be the Banach algebra which is the group algebra of $G$. We consider a variety of Banach left $L^1 (G)$ -modules over $L^1 (G)$ , and seek to determine conditions on $G$ that determine when these modules are either projective or injective or flat in the category. The answers typically involve $G$ being compact or discrete or amenable. For example, in the case where $G$ is discrete and $1 < p < \infty$, we find that the module $\ell^p (G)$ is injective whenever $G$ is amenable, and that, if it is amenable, then $G$ is ‘pseudo-amenable’, a property very close to that of amenability.