Let $\alpha$ be a $\mathbf Z^d$-action ($d\ge 2$) by automorphisms of a compact metric abelian group. For any non-linear shape $I\subset\mathbf Z^d$, there is an $\alpha$ with the property that $I$ is a minimal mixing shape for $\alpha$. The only implications of the form $I$ is a mixing shape for $\alpha$ $\implies$ $J$ is a mixing shape for $\alpha$'' are trivial ones for which $I$ contains a translate of $J$. \par If all shapes are mixing for $\alpha$, then $\alpha$ is mixing of all orders. In contrast to the algebraic case, if $\beta$ is a $\mathbf Z^d$-action by measure-preserving transformations, then all shapes mixing for $\beta$ does not preclude rigidity. \par Finally, we show that mixing of all orders in cones --- a property that coincides with mixing of all orders for $\mathbf Z$--actions --- holds for algebraic mixing $\mathbf Z^2$-actions.