  This paper deals with singularities of closures of $2$-nilpotent Borel
conjugacy classes in either a $\text{GL}_n$-conjugacy class or in the nilpotent
cone of $\text{GL}_n$. In the latter case we construct a resolution of
singularities, in the former we show that singularities are rational by
applying a result of M. Brion. We reason why this generalizes the result of N.
Perrin and E. Smirnov on the rationality of singularities of Springer fiber
components in the two-column case. In the case of Borel orbit closures being
contained in orbital varieties, we give an alternative version of L. Fresse's
recent singularity criterion.
