Is there a way to identify singular points in a spectrahedron without finding the entire set?

Question

I'm a physicist working in a non-linear optimization problem that reduces to a semidefinite program (SDP). In the simple low dimensional cases we workout in the research we could identify singular points which have very interesting physical meaning. However there is higher dimensional cases where we can't see these points and sometimes it's too computationally expensive to find the entire spectrahedron. I looked in a lot of places for geometrical analysis of spectrahedra but found very little material useful for me.