This paper considers the reflection of waves by multiscale interfaces in the framework of the wavelet transform. First, we show how the wavelet transform is efficient to detect and characterize abrupt changes present in a signal. Locally homogeneous abrupt changes have conspicuous cone-like signatures in the wavelet transform from which their regularity may be obtained. Multiscale clusters of nearby singularities produce a hierarchical arrangement of conical patterns where the multiscale structure of the cluster may be identified. Second, the wavelet response is introduced as a natural extension of the wavelet transform when the signal to be analyzed (i.e., the velocity structure of the medium) can only be remotely probed by propagating wavelets into the medium instead of being directly convolved as in the wavelet transform. The reflected waves produced by the incident wavelets onto the reflectors present in the medium constitute the wavelet response. We show that both transforms are equivalent when multiple scattering is neglected and that cone-like features and ridge functions can be recognized in the wavelet response as well. Experimental applications of the acoustical wavelet response show how useful information can be obtained about remote multiscale reflectors. A first experiment implements the synthetic cases discussed before and concerns the characterization of planar reflectors with finite thicknesses. Another experiment concerns the multiscale characterization of a complex interface constituted by the surface of a layer of monodisperse glass beads immersed in water.