  We consider the problem of characterizing the Sobolev wavefront set of a
tempered distribution $u\in\mathcal{S}'(\mathbb{R}^{d})$ in terms of its
continuous wavelet transform, with the latter being defined with respect to a
suitably chosen dilation group $H\subset{\rm GL}(\mathbb{R}^{d})$. We derive
necessary and sufficient criteria for elements of the Sobolev wavefront set,
formulated in terms of the decay behaviour of a given generalized continuous
wavelet transform. It turns out that the characterization of directed
smoothness of finite order can be performed in the two important cases: (1)
bandlimited wavelets, and (2) wavelets with finitely many vanishing moments
(e.g.~compactly supported wavelets).
  The main results of this paper are based on a number of fairly technical
conditions on the dilation group. In order to demonstrate their wide
applicability, we exhibit a large class of generalized shearlet groups in
arbitrary dimensions fulfilling all required conditions, and give estimates of
the involved constants.
