  We study static spherically symmetric solutions to Einstein's equations with
a repulsive singularity at the centre. We show that geodesics are extendible
across the singularity, so the singularity does not lead to pathological
causality properties. It is best described as an irreducible spacetime
boundary. As such it must be assigned to an entropy, so that the total entropy
is a sum of matter entropy and of singularity entropy. We evaluate the latter
by using methods that have been developed for black hole thermodynamics,
namely, Euclidean Quantum Gravity and Wald's Noether charge approach. Then, we
use the maximum-entropy principle in order to show that regular solutions
correspond to global maxima of the total entropy for stellar masses below the
Oppenheimer-Volkoff limit, thus providing a thermodynamic justification to the
regularity assumption employed in all stellar models. The maximum entropy
principle also defines stable singular configurations for masses above the
Oppenheimer-Volkoff limit, which we name singularity stars. We analyse their
properties, and discuss the possibility that they correspond to a new type of
astrophysical object that intermediates between neutron stars and black holes.
