The characterization of the workspace for general spatial 3R chains with skew joint axes is refined by describing the variety of singular displacements as the union of the singular configuration manifolds of orientation type, position type, and attitude type. The surface of attitude singularities is revealed by transferring the singularity sets of spherical 3R chains with intersecting joint axes to the geometry of spatial kinematic chains with skew, non-intersecting joint axes. For this purpose, the degeneracy of a screw system is analyzed by means of a particular angle concept for a set of three oriented lines in space. The obtained argumentation is expressed in terms of geometric manipulator Jacobians completing previous results.