  Information on the phase structure of strongly interacting matter at high
baryon densities can be gained from observations of neutron stars and their
detailed analysis. In the present work Bayesian inference methods are used to
set constraints on the speed of sound in the interior of neutron stars, based
on recent multi-messenger data in combination with limiting conditions from
nuclear physics at low densities. Two general parametric representations are
introduced for the sound speed $c_s$ in order to examine the independence with
respect to choices for the parametrisation of Priors. Credible regions for
neutron star properties are analysed, in particular with reference to the quest
for possible phase transitions in cold dense matter. The evaluation of Bayes
factors implies extreme evidence for a violation of the conformal bound, $c_s^2
\leq 1/3$, inside neutron stars. Given the presently existing data base, it can
be concluded that the occurrence of a first-order phase transition in the core
of even a two-solar-mass neutron star is unlikely, while a continuous crossover
cannot be ruled out. At the same time it is pointed out that the discovery of a
superheavy neutron star with a mass $ M \sim 2.3 - 2.4\, M_\odot$ would
strengthen evidence for a phase change in the deep interior of the star.
