Let $G$ be a complex connected reductive algebraic group. A normal affine $G$-variety $X$ is called spherical if its ring of regular functions $C[X]$ is a multiplicity free $G$-module. A natural question in the study of such varietes is the following: if we only know the $G$-module structure of $C[X]$, how well do we know $X$? Bringing geometry to this question, Alexeev and Brion introduced a certain moduli scheme. After recalling earlier examples of this scheme due to S. Jansou, P. Bravi and S. Cupit-Foutou, I will discuss joint work with S. Papadakis on the case where the module structure of $C[X]$ is that of $C[W]$, where $W$ is a spherical module for a group of type $A$.