  We consider the coorbit theory associated to general continuous wavelet
transforms arising from a square-integrable, irreducible quasi-regular
representation of a semidirect product group $G = \mathbb{R}^d \rtimes H$. The
existence of coorbit spaces for this very general setting has been recently
established, together with concrete vanishing moment criteria for analyzing
vectors and atoms that can be used in the coorbit scheme. These criteria depend
on fairly technical assumptions on the dual action of the dilation group, and
it is one of the chief purposes of this paper to considerably simplify these
assumptions.
  We then proceed to verify the assumptions for large classes of dilation
groups, in particular for all abelian dilation groups, as well as a class
called {\em generalized shearlet dilation groups}, containing and extending all
known examples of shearlet dilation groups employed in dimensions two and
higher. We explain how these groups can be systematically constructed from
certain commutative associative algebras of the same dimension, and give a full
list, up to conjugacy, of shearing groups in dimensions three and four. In the
latter case, three previously unknown groups arise.
  As a result the existence of Banach frames consisting of compactly supported
wavelets, with simultaneous convergence in a whole range of coorbit spaces, is
established for all groups involved.
