The over half-century-old phenomenon of Anderson localization continues to encompass newly emerging physics of wave transport in strongly scattering media. In recent years, localization of classical waves (e.g., light or sound) has been extensively investigated experimentally. Successful experiments in three dimensions used ultrasound to probe disordered aluminum-bead networks (“mesoglasses”) and convincingly demonstrated Anderson localization in these metallic samples. In dielectric materials, on the other hand, it has been predicted that ultrasound localization is prevented by induced long-range electric dipole-dipole interactions, which are screened in metals. Motivated by this prediction, this thesis searches for experimental evidence for this claim by investigating three dielectric mesoglass samples made from weakly-sintered borosilicate glass beads. Resonant frequencies of individual and clusters of beads were measured to study the character of the vibrational modes, and simulations of the modes assisted in visualizing the mechanical deformations. Ballistic and transverse confinement experiments that measure both time- and position-resolved transmission through slab-shaped samples were used to investigate the approach to Anderson localization as a function of ultrasonic frequency. Transmitted ultrasonic intensities and their transverse expansion or confinement were compared with the self-consistent theory of localization (SCT). Although this theory has not been generalized to include electric dipole effects, it is expected to give accurate predictions at short transverse distances ρ where the intensities are large and the weak induced dipolar contributions negligible. For this range of ρ, good fits were obtained, revealing evidence of transport well beyond the mobility edge into localized regimes at some frequencies. By contrast, at large ρ, where dipolar contributions could be significant, the measured transmitted intensities were much larger than SCT predictions. However, the effect of applying a conductive silver coating to screen each bead in one sample was shown experimentally to be ineffective in reducing these anomalous deviations, thereby ruling out the dipolar mechanism as being responsible. A different mechanism is proposed to explain these large ρ anomalies. An explanation is also proposed for why the unique structure of mesoglasses enables a screening mechanism that suppresses long-range electric dipole-dipole interactions, making Anderson localization possible in dielectric mesoglasses, consistent with experimental observations.