  This paper presents a detailed regularity analysis of anisotropic wavelet
frames and subdivision. In the univariate setting, the smoothness of wavelet
frames and subdivision is well understood by means of the matrix approach. In
the multivariate setting, this approach has been extended only to the special
case of isotropic refinement with the dilation matrix all of whose eigenvalues
are equal in the absolute value. The general anisotropic case has resisted to
be fully understood: the matrix approach can determine whether a refinable
function belongs to $C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p <
\infty$, but its H\"older regularity remained mysteriously unattainable.
  It this paper we show how to compute the H\"older regularity in
$C(\mathbb{R}^s)$ or $L_p(\mathbb{R}^s)$, $1 \le p < \infty$. In the
anisotropic case, our expression for the exact H\"older exponent of a refinable
function reflects the impact of the variable moduli of the eigenvalues of the
corresponding dilation matrix. In the isotropic case, our results reduce to the
well-known facts from the literature. We provide an efficient algorithm for
determining the finite set of the restricted transition matrices whose spectral
properties characterize the H\"older exponent of the corresponding refinable
function. We also analyze the higher regularity, the local regularity, the
corresponding moduli of continuity, and the rate of convergence of the
corresponding subdivision schemes. We illustrate our results with several
examples.
