During this talk I will discuss the descriptive set-theoretic complexity of the topological conjugacy relation for free minimal $G$-subshifts for various countable groups $G$. For residually finite countable groups $G$ we will see that there exists a probability measure on the set of free minimal $G$-subshifts, which is invariant under a natural action of $G$ and such that the stabilizers of points in this action are a.e. amenable. As a consequence, we will get that if $G$ is a countable residually finite non-amenable group, then the relation of topological conjugacy on free minimal $G$-subshifts is not amenable. On the other hand, for the group $G=Z$, we will look at the class of regular Toeplitz subshifts and see that the conjugacy relation is an amenable equivalence relation there. This is joint work (in progress) with Todor Tsankov.