  We investigate the existence of embeddings of shearlet coorbit spaces
associated to weighted mixed $L^p$-spaces into classical Sobolev spaces in
dimension three by using the description of coorbit spaces as decomposition
spaces. This different perspective on these spaces enables the application of
embedding results that allow the complete characterization of embeddings for
certain integrability exponents, and thus provides access to a deeper
understanding of the smoothness properties of coorbit spaces, and of the
influence of the choice of shearlet groups on these properties. We give a
detailed analysis, identifying which features of the dilation groups have an
influence on the embedding behavior, and which do not. Our results also allow
to comment on the validity of the interpretation of shearlet coorbit spaces as
smoothness spaces.
