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.