Surface enhanced Raman scattering (SERS) is not restricted to the well-known one-photon excited spontaneous Raman process that gives information on molecular composition, structure, and interaction through vibrational probing with high sensitivity. The enhancement mainly originates in high local fields, specifically those provided by localized surface plasmon resonances of metal nanostructures. High local fields can particularly support nonlinear Raman scattering, as it depends on the fields to higher powers. By revealing plasmon-molecule interactions, nonlinear Raman processes provide a very sensitive access to the properties of metal nanomaterials and their interfaces with molecules and other materials. This Perspective discusses plasmon-enhanced spontaneous and coherent nonlinear Raman scattering with the aim of identifying advantages that lead to an advanced vibrational characterization of such systems. The discussion will highlight the aspects of vibrational information that can be gained based on specific advantages of different incoherent and coherent Raman scattering and their surface enhancement. While the incoherent process of surface enhanced hyper Raman scattering (SEHRS) gives highly selective and spectral information complementary to SERS, the incoherent process of surface enhanced pumped anti-Stokes Raman scattering (SEPARS) can help to infer effective nonresonant SERS cross sections and allows to see "hot" vibrational transitions. Surface enhanced coherent anti-Stokes Raman scattering (SECARS) and surface enhanced stimulated Raman scattering (SESRS) combine the advantages of high local fields and coherence, which gives rise to high detection sensitivity and offers possibilities to explore molecule-plasmon interactions for a comprehensive characterization of composite and hybrid structures in materials research, catalysis, and nanobiophotonics.