  Let either $GL(E)\times SO(F)$ or $GL(E)\times Sp(F)$ act naturally on the
space of matrices $E\otimes F$. There are only finitely many orbits, and the
orbit closures are orthogonal and symplectic generalizations of determinantal
varieties, which can be described similarly using rank conditions. In this
paper, we study the singularities of these varieties and describe their
defining equations. We prove that in the symplectic case, the orbit closures
are normal with good filtrations, and in characteristic $0$ have rational
singularities. In the orthogonal case we show that most orbit closures will
have the same properties, and determine precisely the exceptions to this.
