Unlike parallel robots, for which hundreds of different architectures have been proposed, the vast majority of six-degree-of-freedom (DOF) serial robots have one of two simple architectures. In both architectures, the inverse kinematics can be solved in closed form and the singularities described by trivial geometric and algebraic conditions. These conditions can be readily obtained by analyzing the determinant of the robot’s Jacobian matrix, and provide an in-depth understanding of the robot’s singularities, which is essential for its optimal use. However, for various reasons, robot arms with unorthodox architectures are occasionally designed. Such arms do not have closed-form inverse kinematics and little insight into their singularities can be gained by analyzing the determinant of their Jacobian. One such robot arm for which the conventional singularity analysis approach fails is the new Link 6 collaborative robot by Kinova. In this paper, we study the complex singularities of Link 6 by investigating all possibilities for screw dependencies, deriving a simple equation for each case, and then describing each singularity type using Grassmann line geometry. Twelve different singularity configurations are identified and described with seven relatively simple geometric conditions. Our approach is general and can be applied to other robot arms.