Coorbit spaces are families of Banach spaces of functions defined in terms of their coefficient decay, associated with a wavelet system constructed using a square-integrable representation of a locally compact group. Many established function spaces, such as Besov spaces, shearlet coorbit spaces, Sobolev–Shubin spaces, and modulation spaces, are fundamental examples of this construction. In this talk, I will discuss Feichtinger–Grochenig theory of the construction of these spaces and discuss some of its properties. As an example, we talk about wavelet coorbit spaces. We seek to achieve a comprehensive classification of generalized wavelet coorbit spaces in dimension two, i.e., the function spaces associated with the generalized wavelet systems arise from the quasi-regular representation to semidirect product groups in dimension two, based on their approximation-theoretic properties. The theory is generally well-established by now, at least as far as the study of a single such scale is concerned. Comparing these scales for two fundamentally different generalized wavelet systems is a fundamental problem in this area that is only partially understood. We aim to provide a complete answer to this question for this particular subclass of generalized wavelet systems in dimension two. This is based on a joint work with Hartmut Fuhr and Noufal Asharaf.