  The rationality of the singularities of the $A_n$-loci is the natural
question that arises in the papers devoted to the study of the Thom polynomials
and $K$-theoretic invariants of the said loci. In this paper we prove that, in
general, the $A_2$-loci have singularities worse than rational, and therefore
they can not be presented as a GIT quotient of a smooth variety with respect to
a reductive group.
