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.