In recent years directional multiscale transformations like the curvelet- or
shearlet transformation have gained considerable attention. The reason for this
is that these transforms are, unlike more traditional transforms like wavelets,
able to efficiently handle data with features along edges. The main result
confirming this property for shearlets is contained in [G. Kutyniok, D. Labate.
Resolution of the Wavefront Set using continuous Shearlets, Trans. AMS 361
(2009), 2719-2754] where it is shown that for very special functions $\psi$
with frequency support in a compact conical wegde the decay rate of the
shearlet coefficients of a tempered distribution $f$ with respect to the
shearlet $\psi$ can resolve the Wavefront Set of $f$. We show an analogous
result where the only requirement we impose on $\psi$ is essentially to possess
sufficiently many anisotropic vanishing moments. We also show how to build
frames for $L^2(R^2)$ from any such function.