Understanding the characteristics of electron beam bombardment that induce electrostatic discharge (ESD) of insulating materials is crucial to constructing an electrically stable spacecraft. A measurement system has been designed to determine the beam energy and charge flux densities at which typical spacecraft materials intended for the James Webb Space Telescope (JWST) undergo ESD. Because discharge events occur over time intervals ranging from nanoseconds to minutes, multiple detection methods were employed as charge was accumulated on a sample surface; these methods included monitoring of sample current and optical emissions from the sample surface. Each sample was also examined with optical microscopy before and after testing to determine permanent changes in the materials. Testing for various samples was done at 130 K, 170 K, and 200 K. An overview of ESD incidence for all materials is provided in relation to beam conditions and material composition. Data for a carbon fiber material is discussed in greater detail, comparing results from each detection method. Two discharge modes were observed: a sudden-onset, long-duration exponentially-decaying sample current accompanied by the release of intense blue photons, and a sudden spike or arc in current with a white light flash. The implications of test results for JWST are discussed.