Suppose we have an $n$-DOF robot manipulator and let $q \in \mathbf{R}^n$ denotes a robot joint configuration. Then a singular configuration $q'$ is a configuration at which the Jacobian $J(q')$ does not have maximum rank. Let $S$ be the set of all singular configurations of a given robot
Is there any (general) result regarding characterization of the set $S$? Any work discussing or answering questions such as "Is $S$ a manifold?", "Does $S$ contain only isolated points?", etc.?
So far I have found quite a few work talking about classification of singular configurations. I think they still do not really answer my questions. Can anyone point me to some related stuff?