Abstract <p>Periodic flows propagating along the interface between two viscous, stratified, weakly compressible fluids are analyzed. Analytical complete asymptotic solutions of the dispersion equations are constructed using the methods of the singular perturbation theory, which determine both the large-scale waves and the fine-structured ligamental components of the flow. It is shown that an additional class of singular solutions arises in the three-dimensional problem and disappears in the two-dimensional one; this class describes the additional ligament component of the periodic flow.</p>