Spontaneous vibrational Raman scattering is a ubiquitous form of light-matter interaction whose description necessitates quantization of the electromagnetic field. It is usually considered an incoherent process since it stems from the fluctuations of the vacuum field, and hence the scattered field lacks any predictable phase relationship with the incoming one. Consequently, spontaneous Raman scattering is often described in the single-molecule picture, the total signal being the incoherent sum of random single-molecule events. Nevertheless, basic considerations suggest that this common picture may not always be valid. For one, the Stokes photon and the quantum of vibration (the phonon) are emitted in correlated pairs. Second, the diffraction limit sets a fundamental bound on the spatial information that is in principle measurable about which molecule got excited. We may therefore inquire: What is the quantum state of the vibrating matter after spontaneous scattering? Does it depend on the measurement apparatus? Since the intensity of anti-Stokes scattering is highly sensitive to the collective vibrational state, can we learn about vibrational coherence from the temporal and spectral evolution of the Stokes--anti-Stokes correlations? Can these correlations manifest some form of entanglement when other four-wave mixing channels may interfere? In this thesis, we experimentally address these questions. We investigate the relationship between optical and quantum coherence and show that the latter may emerge even in the spontaneous regime as a result of a selection process. We first measure time-resolved Stokes--anti-Stokes correlations on a molecular liquid sample. We implement a detection scheme that erases the which-molecule information in space and time, so that we prepare the vibrational state after spontaneous scattering in a coherent collective quantum superposition and record the same hallmarks of vibrational coherence as would be observed in stimulated Raman spectroscopy. Next, working with a crystalline sample, we use fiber dispersion spectroscopy and fast single-photon detectors to spectrally resolve the Stokes--anti-Stokes coincidences. With this technique, we demonstrate that the quantum interference between electronic and vibrational contributions to the $\chi^{(3)}$ tensor of diamond generates broadband polarization-entangled photon pairs. We characterize this entanglement through a Bell test. Finally, we show that the spontaneous Raman scattered signal can be used as a proxy to measure how the near-field of a photonic resonator couples to the far field. We record confocal maps of Raman scattering from organic molecules embedded in plasmonic nanocavities; by employing different cylindrical vector beams, these measurements reveal the polarization of the excited antenna modes, offering a novel approach to near-field probing. The techniques we develop in this thesis can be applied to various and more complex material systems, paving the way for further fundamental studies on the coherence properties of light and matter, and for technological applications such as integrated photonics for quantum information and nanodevices based on ultraconfined light modes.