To understand the nuclear stellar populations and star formation histories of the nuclei of spiral galaxies, we have obtained K-band nuclear spectra for 41 galaxies and H-band spectra for 20 galaxies in the Infrared Space Observatory's Atlas of Bright Spiral Galaxies. In the vast majority of the subsample (80%), the near-infrared spectra suggest that evolved red stars completely dominate the nuclear stellar populations and that hot young stars are virtually nonexistent. The signatures of recent star formation activity are only found in 20% of the subsample, even though older red stars still dominate the stellar populations in these galaxies. Given the dominance of evolved stars in most galaxy nuclei and the nature of the emission lines in the galaxies where they were detected, we suggest that nuclear star formation proceeds in the form of instantaneous bursts. The stars produced by these bursts comprise only ~2% of the total nuclear stellar mass in these galaxies, but we demonstrate how the nuclear stellar populations of normal spiral galaxies can be built up through a series of these bursts. The bursts were detected only in Sbc galaxies and later, and both bars and interactions appeared to be sufficient, but not necessary, triggers for the nuclear star formation activity. The vast majority of galaxies with nuclear star formation were classified as H II galaxies. With one exception, LINERs and transition objects were dominated by older red stars, which suggested that star formation was not responsible for generating these galaxies' optical line emission.