We define the singularity degree of a face which is not necessarily facially exposed. We show that the singularity degree of a linear conic optimization problem is equal to the singularity degree of the minimal face on the linear image of the convex cone, which generalises the result by Drusvyatskiy, Pataki and Wolkowicz in 2015. Given pairwise distances of a framework (G, p), the singularity degree of the corresponding EDM completion problem is related to the stability of the framework (high singularity means the framework is less stable). As an application, we give an upper bound of the singularity degree for generic frameworks and tensegrities underlying a Laman plus d graph ( Laman graph plus d edges). This is joint work with Henry Wolkowicz.