Vertically-articulated six-axis serial robots exhibit three types of singularities: wrist singularities, shoulder singularities, and elbow singularities. A wrist singularity occurs when axes 4 and 6 are coincident. A shoulder singularity occurs when the wrist center lies on a cylinder centered about axis 1 and with a radius equal to the distance between axes 1 and 4. Finally, an elbow singularity occurs when the wrist center lies in the same plane as axes 2 and 3.
When a robot is controlled in cartesian mode and passes near a singularity, the speed of some joints becomes suddenly very large and the cartesian velocity of the end-effector is significantly reduced. For this reason, trajectories that make the robot pass near singular configurations should be avoided.
The following video illulstrates the three types of singularities:
An excellent reference on this topic is the following scientific paper:
M.J.D. Hayes, M.L. Husty, P.J. Zsombor-Murray, 2002, “Singular Configurations of Wrist-Partitioned 6R Serial Robots: a Geometric Perspective for Users”, Transactions of the Canadian Society for Mechanical Engineering, Vol. 26, No. 1, pp. 41–55.