In the recent years, certain wavelet-type transformations such as the curvelet or shearlet transformation have gained considerable attention, due to their potential for efficiently handle data with features along edges. Namely in both cases, it was shown that the decay rate of the corresponding transformation coefficients of a tempered distribution can resolve the wavefront set of the distribution. Roughly speaking, the wavefront set of a tempered distribution f is the set of points t ∈ Rn and directions ξ ∈ S^(n-1) along which f is not smooth at t. Recently, many efforts have been made aiming to generalize the above characterization, i.e. characterization of the wavefront set of a tempered distribution in terms of its continuous wavelet transform, for higher dimensional continuous wavelet. In this talk, we consider the problem of characterizing the Sobolev wavefront set of a distribution for a higher-dimensional wavelet transform in two important cases where: 1) the mother wavelet is compactly supported, and 2) the mother wavelet has compactly supported Fourier transform. This talk is based on joint work with Hartmut Fuhr.