The presence of mesoscopic features and damage in quasi-brittle materials causes significant second-order and nonlinear effects on the acoustic wave propagation characteristics. In order to quantify the influence of such micro-inhomogeneities, a new and promising tool for nondestructive material testing has been developed and applied in the field of damage detection. The technique focuses on the acoustic nonlinear (i.e., amplitude-dependent) response of one of the material’s resonance modes when driven at relatively small wave amplitudes. The method is termed single-mode nonlinear resonance acoustic spectroscopy (SIMONRAS). The behavior of damaged materials is manifested by amplitude dependent resonance frequency shifts, harmonic generation, and nonlinear attenuation. We illustrate the method by experiments on artificial slate tiles used in roofing construction. The sensitivity of this method to discern material damage is far greater than that of linear acoustic methods.