  We revisit the general topic of thermodynamical stability and critical
phenomena in black hole physics, analyzing in detail the phase diagram of the
five dimensional rotating black hole and the black rings discovered by Emparan
and Reall. First we address the issue of microcanonical stability of these
spacetimes and its relation to thermodynamics by using the so-called Poincare
(or "turning point") method, which we review in detail. We are able to prove
that one of the black ring branches is always locally unstable, showing that
there is a change of stability at the point where the two black ring branches
meet. Next we study divergence of fluctuations, the geometry of the
thermodynamic state space (Ruppeiner geometry) and compute the appropriate
critical exponents and verify the scaling laws familiar from RG theory in
statistical mechanics. We find that, at extremality, the behaviour of the
system is formally very similar to a second order phase transition.
