Most of the current commercial collaborative robots present a non-spherical wrist, so they cannot utilize singularity handling techniques efficiently to avoid excessive safety stops while dynamically avoiding collisions. These robots usually require heavier algorithms due to their kinematics or online methods that shift the original singularities. Therefore, to enable more efficient computations on singularity handling and collision avoidance controllers, this manuscript proposes a novel method to characterize singular configurations of non-spherical wrist collaborative robots (6 and 7 Degrees of Freedom). This method is based on a new decoupled kinematic model that allows lighter kinematic computations and enables the joint-dependant characterization of the robot singularities to avoid shifting the singular configurations. Finally, the proposed kinematic model is particularized for a UR10e, where its kinematic behavior has been tested against two different literature models in simulation. In this manner, a novel singularity category (belonging to the internal singularities) is proposed, and a new closed set of characterized singular solutions is obtained.