  We are interested in studying the stationary solutions and phase transitions
of aggregation equations with degenerate diffusion of porous medium-type, with
exponent $1 < m < \infty$. We first prove the existence of possibly infinitely
many bifurcations from the spatially homogeneous steady state. We then focus
our attention on the associated free energy proving existence of minimisers and
even uniqueness for sufficiently weak interactions. In the absence of
uniqueness, we show that the system exhibits phase transitions: we classify
values of $m$ and interaction potentials $W$ for which these phase transitions
are continuous or discontinuous. Finally, we comment on the limit $m \to
\infty$ and the influence that the presence of a phase transition has on this
limit.
