While structures continue to shrink in size for novel optoelectronic applications, the ability to study new optical phenomenon on the nanoscale becomes important. Efforts in this area face challenges as conventional optical techniques fail to achieve the necessary spatial resolution due to fundamental diffraction limitations. A unique and powerful solution for performing near-field optical spectroscopy with nanometer spatial resolution using relativistic electrons in a scanning transmission electron microscope (STEM) and electron energy loss spectroscopy (EELS) environment is presented. Relativistic electrons can be focused down to nanometer-sized probes, and can behave as a virtual, broad-band source of light. By taking advantage of these properties, the photonic density of states of far-ultraviolet whispering gallery modes in SiO2 nanospheres are excited by passing an electron beam close to the sphere. In a similar manner, the waveguide modes of GaN and Ge nanowires are measured, where the nanowire is oriented perpendicular to the electron optical axis. For the Ge nanowires, a multi-mode and zero-mode system is probed. A dielectric formalism describing the energy loss of the electron due to the nanowire is developed in a relativistic framework, and shows quantitative agreement with experimental results. Calculations of nanowire shell systems consisting of a dielectric core and metallic shell are also presented, where the interaction between the surface plasmonic modes and the dielectric waveguide modes are observed. The results open promising gateways for spatially-resolved optical observations of more complex nanostructures.