This paper discusses the criteria for both global Hadamard and dissipative stabilities of viscoelastic constitutive equations (CEs) with instantaneous elasticity. In order to formulate as generally as possible the conditions of stability analyses, we used some unified forms for CEs of differential Maxwell-like and factorable single integral types. The obtained criteria formulate the constraints which have to be imposed on parameters or functions in CEs to avoid the unphysical instabilities in the whole range of Deborah numbers, independent of the flow situation. The problem of Hadamard stability for the viscoelastic CEs is reduced to that well known in the theory of nonlinear elasticity where the complete solution of the problem was found recently in an algebraic form. The criteria of dissipative stability, which have been established previously, are also discussed. When applied to many popular viscoelastic models, the combined stability criteria impose a set of such severe constraints that no single integral CE with time-strain separability proposed in the literature, could satisfy them, and only few differential models have proved to be stable.