Parallel mechanisms frequently contain an unstable type of singularity that has no counterpart in serial mechanisms. When the mechanism is at or near this type of singularity, it loses the ability to counteract external forces in certain directions. The determination of unstable singular configurations in parallel robots is challenging in general, and is usually tackled via an exhaustive search of the workspace using an accurate analytical model of the mechanism kinematics. This paper considers the singularity determination problem from a geometric perspective for planar and spatial three-legged parallel mechanisms. By using the constraints on the passive joint velocities, we derive a necessary condition for the unstable singularities. Using this condition, certain singularities can be found for certain type of platforms. As an example, new singular poses are discovered using this approach for a six-degree-of-freedom machining center.