  We explore classifiability of crossed products of actions of countable
amenable groups on compact, metrizable spaces. It is completely understood when
such crossed products are simple, separable, unital, nuclear and satisfy the
UCT: these properties are equivalent to the combination of minimality and
topological freeness, and the challenge in this context is establishing
$\mathcal{Z}$-stability. While most of the existing results in this direction
assume freeness of the action, there exist numerous natural examples of
minimal, topologically free (but not free) actions whose crossed products are
classifiable.
  In this work, we take the first steps towards a systematic study of
$\mathcal{Z}$-stability for crossed products beyond the free case, extending
the available machinery around the small boundary property and almost
finiteness to a more general setting. Among others, for actions of groups of
polynomial growth with the small boundary property, we show that minimality and
topological freeness are not just necessary, but also \emph{sufficient}
conditions for classifiability of the crossed product.
  Our most general results apply to actions that are essentially free, a
property weaker than freeness but stronger than topological freeness in the
minimal setting. Very recently, M. Joseph produced the first examples of
minimal actions of amenable groups which are topologically free and not
essentially free. While the current machinery does not give any information for
his examples, we develop ad-hoc methods to show that his actions have
classifiable crossed products.
