<p>The aim of this study is to develop and apply a semi-analytical approach to determine the critical parameters of stability loss under compression of a bilayer consisting of a thin film on a semi-infinite substrate in the presence of an interface delamination of a given length. Using the methods of three-dimensional linearized stability theory, the corresponding boundary value problem is reduced to an eigenvalue problem with respect to the parameters of the compressive loading for a system of Fredholm integral equations of the first kind, which is solved numerically. Since the components of the bilayer are considered as hyperelastic materials with an arbitrary form of elastic potential, the obtained results cover a wide range of mechanical characteristics – both in the small and finite (large) deformation modes. Based on examples involving specific types of elastic potentials, the influence of the constitutive model on the critical parameters of stability loss is investigated. For cases of relatively short and relatively long delaminations, a comparison of the results obtained using the proposed approach is made with the results obtained using known approximate approaches used to estimate critical wrinkling and buckling deformations, respectively. In the case of intermediate-length delaminations, it is shown that due to the interaction of wrinkling and buckling mechanisms, the critical values of relative compression in the bilayer are significantly lower than those predicted by the approximate models. At the same time, the obtained results are in good agreement with the results of numerical simulations reported in previous studies and, thus, can be used to predict the onset of interfacial failure in compressed bilayers with thin films.</p>