Is classification of a pair of linear operators $A$, $B$, $A^2=B^2=0$, up to simultaneous conjugation a "wild" problem? Reference?
Equivalently, is it possible to classify finite-dimensional representations of $k\langle x,y \rangle /\langle x^2,y^2 \rangle $, up to equivalence?
It’s tame.
This is proved by Bondarenko in “Representations of dihedral groups over a field of characteristic two”, Math USSR, Sbornik, vol 25, no. 1, (1975) 58-68.
http://iopscience.iop.org/article/10.1070/SM1975v025n01ABEH002197/meta
This result works in any characteristic. It’s only the applications to dihedral groups that require characteristic two.