One of the main challenges faced by the structural health monitoring community is acquiring and processing huge sets of acoustic wavefield data collected from sensors, such as scanning laser Doppler vibrometers or ultrasonic scanners. In fact, extracting information that allows the estimation of the damage condition of a structure can be a time-consuming process. This paper presents a damage detection and localization technique based on a compressive sensing algorithm, which significantly allows us to reduce the acquisition time without losing in detection accuracy. The proposed technique exploits the sparsity of the wavefield in different representation domains, such as those spanned by wave atoms, curvelets, and Fourier exponentials to recover the full wavefield and, at the same time, to infer the damage location, based on comparison between the wavefield reconstructions produced by the different representation domains. The procedure is applied to three different setups related to an aluminum plate with a notch, a glass fiber reinforced polymer plate with a notch, and a composite plate with a delamination. The results show that the technique can be applied in a variety of structural components to reduce acquisition time and achieve high performance in defect detection and localization by removing up to 80% of the Nyquist sampling grid.