An elastic waveguide consists of a straight strip covered with a thin (ℎ ≪ 1) periodic coating at one of strip’s lateral surfaces. The material of the coating is much softer than the one of the massif but their densities are similar. Under a certain relationship between the physical and geometric parameters of the composite waveguide, an asymptotic analysis as ℎ → +0 demonstrates the effect of plurality of spectral gaps, i.e. stopping zones for elastic waves. Moreover, local perturbations of the waveguide profile can bring into the discrete spectrum eigenvalues either below the essential spectrum, or inside discovered gaps. In other words, matching certain parameters in a periodic composite elastic waveguide provides any prescribed number of open gaps in the spectrum as well as any prescribed number of isolated eigenvalues in these gaps and the corresponding trapped modes. Both, travelling and trapped waves at frequencies in the spectral bands, passing zones, and in the discrete spectrum, respectively, provoke localization and concentration of shear stresses at the interface near points where the coating profile function attends its maxima so that the fracture process can be predicted in the vicinity of these points and realizes as fragmentation of the adhesive and a sparsely distributed exfoliation of the thin light periodic coating.