Is the classification of supercuspidal representations a wild problem? As explained by Torsten Ekedahl in this reply, the problem of classifying the irreducible (infinite dimensional) representations of $sl_2(\mathbb{C})$ is a wild one. The notion of wild classification problem was discussed in this question. It means that the problem in a sense contains the problem of reducing pairs of matrices over a field to a canonical form by simultaneous conjugation.

Let $F$ be a non-Archimedean local field. Is the classification of supercuspidal representations of $\mathrm{GL}_n(F)$ a wild classification problem?

The work of Bushnell and Kutzko [The admissible dual of $\mathrm{GL}(N)$ via compact open subgroups] gives a classification of the supercuspidal representations of $\mathrm{GL}_n(F)$, and the local Langlands correspondence for $\mathrm{GL}_n$ also gives a parametrisation of the supercuspidal representations. However, it is not obvious to me whether any of these classifications answers the question.
 A: 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. 
