rt.representation theory - Is the classification of supercuspidal representations a wild problem?

I cannot directly answer your question because I am not a specialist of wild classification problems. However I want to make the following remark that could help.

I was told that the problem of classifying the irreducible (smooth) complex representations of ${rm GL}_n ({mathfrak o}_F )$, where ${mathfrak o}_F$ is the ring of integers of a local field $F$, is wild (at least for $n$ large).

Moreover to construct and classify supercuspidal representations, Bushnell and kutzko obtain them as compactly induced representations from irreducible smooth representations of compact mod center subgroups. In particular we obtain a large class of supercuspidals by inducing certain irreducible complex representations of $F^{times}{rm GL}_n ({mathfrak o}_F)$.

However in Bushnell and Kutzko's theory not all representations of ${rm GL}_n ({mathfrak o}_F )$ are needed, but only very particular ones and one knows how to construct them effectively. Indeed the whole Bushnell and Kutzko's construction is based on the theory of 'simple characters' which is in principal entirely effective.