<p>A plane problem of semibounded piecewise-homogeneous body compression along a straight media interface is investigated in the context of the three-dimensional linearized stability theory. The substrate and coating materials are considered hyperelastic, with arbitrary structures for their elastic potentials. Two characteristic cases of component contact are examined: a rigid, flawless connection along the entire component interface and a finite interfacial crack with unstressed edges. For determining the critical load parameters that induce surface instability, the initial boundary-value problem is reduced to a transcendental equation in the first case and to an eigenvalue problem for a system of Fredholm integral equations of the first kind in the second case. The numerical results obtained for the materials with a harmonic elastic potential show that their compressibility can significantly affect the critical compression parameters, which should be given proper weight in design engineering.</p>