Raman scattering is a powerful probe of local bonding, strain, temperature, and other properties of materials via their influence on vibrational modes or optical phonons. Tip-enhanced Raman spectroscopy (TERS), in which plasmonic modes are excited at the apex of a metal-coated scanning probe tip, enables Raman scattering signals to be detected from nanoscale volumes with precise positional control. We discuss the application of TERS to characterize a variety of semiconductor nanostructures. In studies of Ge-SiGe core-shell nanowires, we measure spatially resolved Raman spectra along the length of a tapered nanowire to demonstrate the ability to measure local strain distributions with nanoscale spatial resolution. In tip-induced resonant Raman spectroscopy of monolayer and bilayer MoS2, we observe large enhancements in Raman signal levels measured for MoS2 associated with excitation of plasmonic gap modes between an Au-coated probe tip and Au substrate surface onto which MoS2 has been transferred. Transitions in B exciton photoluminescence intensity between monolayer and bilayer regions of MoS2 are observed and discussed. Significant differences in nanoscale Raman spectra between monolayer and bilayer MoS2 are also observed. The origins of specific resonant Raman peaks, their dependence on MoS2 layer thickness, and spatial resolution associated with the transition in Raman spectra between monolayer and bilayer regions are described.