<p>Using general solutions of the equilibrium equations of linearized stability theory, transcendental equations for determining the critical strains corresponding to the onset of wrinkling of a thin coating film located on a semibounded substrate are obtained. The substrate/film (bilayer) system is assumed to be under plane strain conditions, and the body materials are nonlinearly elastic with arbitrary structures of elastic potentials. Two variants of the boundary conditions at the interface are considered: perfectly bonded layers and perfectly lubricated layers, corresponding to the “strongest” and “weakest” types of bonding between the bilayer components. Numerical results for determining the critical values of the wrinkling strain are presented for the harmonic potential (compressible bodies, large strains), the quadratic potential (compressible bodies, small strains), the Treloar potential, and the Bartenev–Khazanovich potential (incompressible bodies, large strains). The nature of the dependence of the critical strain and critical wavelength on the elastic constants of the substrate and film materials and on the type of elastic potential was studied. A comparison of the obtained results with known theoretical and experimental results was carried out.</p>