Shackleton crater at the Moon's south pole has been suggested as a possible site of concentrated deposits of water ice, on the basis of modelling of bi-static radar polarization properties and interpretations of earlier Earth-based radar images. This suggestion, and parallel assumptions about other topographic cold traps, is a significant element in planning for future lunar landings. Hydrogen enhancements have been identified in the polar regions, but these data do not identify the host species or its local distribution. The earlier Earth-based radar data lack the resolution and coverage for detailed studies of the relationship between radar scattering properties, cold traps in permanently shadowed areas, and local terrain features such as the walls and ejecta of small craters. Here we present new 20-m resolution, 13-cm-wavelength radar images that show no evidence for concentrated deposits of water ice in Shackleton crater or elsewhere at the south pole. The polarization properties normally associated with reflections from icy surfaces in the Solar System were found at all the observed latitudes and are strongly correlated with the rock-strewn walls and ejecta of young craters, including the inner wall of Shackleton. There is no correlation between the polarization properties and the degree of solar illumination. If the hydrogen enhancement observed by the Lunar Prospector orbiter indicates the presence of water ice, then our data are consistent with the ice being present only as disseminated grains in the lunar regolith.