  We study continuous wavelet transforms associated to matrix dilation groups
giving rise to an irreducible square-integrable quasi-regular representation on
${\rm L}^2(\mathbb{R}^d)$. We first prove that these representations are
integrable as well, with respect to a wide variety of weights, thus allowing to
consistently quantify coefficient decay via coorbit space norms. We then show
that these spaces always admit an atomic decomposition in terms of bandlimited
Schwartz wavelets. We exhibit spaces of Schwartz functions densely contained in
(most of) the coorbit spaces. We also present an example showing that for a
consistent definition of coorbit spaces, the irreducibility requirement cannot
be easily dispensed with.
  We then address the question how to predict wavelet coefficient decay from
vanishing moment assumptions. To this end, we introduce a new condition on the
open dual orbit associated to a dilation group: If the orbit is temperately
embedded, it is possible to derive rather general weighted ${\rm
L}^{p,q}$-estimates for the wavelet coefficients from vanishing moment
conditions on the wavelet and the analyzed function. These estimates have
various applications: They provide very explicit admissibility conditions for
wavelets and integrable vectors, as well as sufficient criteria for membership
in coorbit spaces. As a further consequence, one obtains a transparent way of
identifying elements of coorbit spaces with certain (cosets of) tempered
distributions.
  We then show that, for every dilation group in dimension two, the associated
dual orbit is temperately embedded. In particular, the general results derived
in this paper apply to the shearlet group and its associated family of coorbit
spaces, where they complement and generalize the known results.
