It is now possible to establish the chemical identity of a substance at the ultimate detection limit of a single molecule, i.e. the sensitivity required to probe 1.66 yoctomoles (1/NA), using surface-enhanced Raman scattering (SERS). It is also possible to image within an individual molecule, all while retaining chemical selectivity, using tip-enhanced Raman scattering (TERS). The potential applications of ultrasensitive SERS and TERS in chemical and biological detection and imaging are evident, and have attracted significant attention over the past decade. Rather than focusing on conventional single/few-molecule SERS and TERS experiments, where the objective is ultrasensitive spectroscopy and nanoscale chemical imaging, herein we consider greatly informative, yet significantly underrated, signatures of single molecules. Namely, we review recent efforts ultimately aimed at probing different aspects of a molecule's local environment through a detailed analysis of its SERS and TERS spectra and images. Particular attention is devoted to local electric field imaging using TERS; we describe how the vector components and absolute magnitude of the local electric field may be inferred from molecular Raman signatures. We also propose experiments that can potentially be used to cross-check the insights gained from the described SERS and TERS measurements. The ultimate goal of this review is to demonstrate that there is much more to single-molecule Raman scattering than mere ultrasensitive chemical analysis.