  The distribution of dark matter around galactic or cluster halos has usually
been assumed to be approximately isothermal with a non-zero core radius, which
is expected to be of the order of the size of the visible matter distribution.
Recently, the possibility has been raised that dark matter halos might be
singular in the sense that the dark matter density $\rho$ could increase
monotonically with radius $r$ down to a very small distance from the center of
galaxies or clusters. Such central cusps in the dark matter density could lead
to a high flux of gamma rays from WIMP dark matter annihilation. Here we
analyze two possibilities that have been discussed in the literature, $\rho
\propto r^{-n}$ with $n \approx 1\ {\rm or}\ 2$, and point out that such
density profiles are excluded by gravitional lensing analyses on cluster scales
and by the rotation curves of gas-rich, halo-dominated dwarf spirals on small
scales. We also point out that if spiral galaxies form by gas infall inside
dark matter halos, as they are expected to do in any hierarchical clustering
model, such profiles almost always lead to falling rotation curves after
infall, contrary to observations.
