  This paper presents a full catalogue, up to conjugacy and subgroups of finite
index, of all matrix groups $H < {\rm GL}(3,\mathbb{R})$ that give rise to a
continuous wavelet transform with associated irreducible quasi-regular
representation. For each group in this class, coorbit theory allows to
consistently define spaces of sparse signals, and to construct atomic
decompositions converging simultaneously in a whole range of these spaces. As
an application of the classification, we investigate the existence of compactly
supported admissible vectors and atoms for the groups.
