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Sobolev wavefront sets and $2$-microlocal spaces play a key role indescribing and analyzing the singularities of distributions in microlocalanalysis and solutions of partial differential equations. Employing thecontinuous shearlet transform to Sobolev spaces, in this paper we characterizethe microlocal Sobolev wavefront sets, the $2$-microlocal spaces, and localH\"older spaces of distributions/functions. We then establish the connectionsamong Sobolev wavefront sets, $2$-microlocal spaces, and local H\"older spacesthrough the continuous shearlet transform.

Microlocal analysis and characterization of Sobolev wavefront sets using shearlets