The solar wind experiment on the Advanced Composition Explorer (ACE) has detected at least 328 impulsive solar electron bursts at energies below 1.4 keV that exhibited energy and angle dispersion in the interval from 1 January 1998 through 30 September 2003. Some of these bursts could be detected down to energies as low as 73 eV, the lowest energies yet reported for such events. The study interval encompassed the rise to and initial decline from the most recent solar activity maximum. We find that the average rate of occurrence of these electron bursts, which were strongly associated with intense type III radio bursts that extended down close to the local plasma frequency, did not vary significantly as a function of solar activity as measured by smoothed monthly sunspot numbers. The bursts were observed within all types of solar wind flows, including recurrent low-speed and high-speed solar wind and transient disturbances driven by coronal mass ejections; however, the bursts were observed predominantly within low-density solar wind, particularly those bursts that could be detected at the very lowest energies. This preferred association with low-density solar wind suggests the possibility that energy losses to Coulomb collisions in the very inner heliosphere play a significant role in determining which low-energy burst electrons are transmitted to 1 AU. Few low-energy electron bursts were observed in the near-vicinity of the heliospheric current sheet, which indicates that if the bursts are a result of magnetic reconnection in the solar atmosphere the reconnection must usually occur at current sheets that do not extend out into the solar wind. The occurrence of some of the more intense bursts within recurrent high-speed streams, which are thought to originate from coronal holes or their boundary regions, is something of a puzzle. We suggest four possible explanations of this puzzle.