Shearlet transforms have been introduced as class of directionally selective wavelet transforms. One way of describing the approximation-theoretic properties of such generalized wavelet systems relies on coorbit spaces, i.e., spaces defined in terms of sparsity properties with respect to the system. In higher dimensions, there are several distinct possibilities for the definition of shearlet systems, and their approximation-theoretic properties are currently not well-understood. In this note we investigate shearlet systems in dimension three. Here there are basically two distinct types of shearing available, and we want to clarify whether the resulting coorbit spaces are distinct. The analysis of these spaces relies on an alternative description via decomposition spaces, and a recently developed, rather comprehensive theory that allows to decide inclusion relationships of decomposition spaces in a computational fashion. These results can be employed to clarify the relationship of coorbit spaces defined over different groups, as well as the relationship of coorbit spaces to classical smoothness spaces such as Sobolev spaces. We show that different shearlet dilation groups in dimension three indeed give rise to different scales of coorbit spaces.
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