  We investigate the properties of a special class of singular solutions for a
self-gravitating perfect fluid in general relativity: the singular isothermal
sphere. For arbitrary constant equation-of-state parameter $w=p/\rho$, there
exist static, spherically-symmetric solutions with density profile $\propto
1/r^2$, with the constant of proportionality fixed to be a special function of
$w$. Like black holes, singular isothermal spheres possess a fixed
mass-to-radius ratio independent of size, but no horizon cloaking the curvature
singularity at $r=0$. For $w=1$, these solutions can be constructed from a
homogeneous dilaton background, where the metric spontaneously breaks spatial
homogeneity. We study the perturbative structure of these solutions, finding
the radial modes and tidal Love numbers, and also find interesting properties
in the geodesic structure of this geometry. Finally, connections are discussed
between these geometries and dark matter profiles, the double copy, and
holographic entropy, as well as how the swampland distance conjecture can
obscure the naked singularity.
