<p>This paper investigates the cone of copositive matrices that together with its dual, the completely positive cone, has garnered significant attention due to their diverse applications. Despite their importance, the intricate geometric properties of these cones, particularly their facial structures, remain only partially understood. A key aspect of studying cones, and matrix cones in particular, lies in analyzing their boundary structure, defined by their faces. This work aims to examine the singularity degrees of the faces of the copositive cone, which can be considered as a measure of their complexity.</p>