  One-dimensional Bose gases with contact repulsive interactions are
characterized by the presence of infinite-lifetime quasiparticles whose momenta
are called the `rapidities'. Here we develop a probe of the local rapidity
distribution, based on the fact that rapidities are the asymptotic momenta of
the particles after a long one-dimensional expansion. This is done by
performing an expansion of a selected slice of the gas. We first apply this
idea to a cloud in the quasi-condensate regime at equilibrium in a trap. We
obtain an experimental picture of the position-dependent rapidity distribution
which is in fair agreement with the theory prediction. The asymptotic regime is
barely reached, but we show that finite expansion time can be taken into
account using the Generalized Hydrodynamics theory. We then apply this local
probe to an out-of-equilibrium situation where the local rapidity distribution
is expected to be doubly peaked -- a hallmark of a non-thermal state -- even
though the global rapidity distribution would possess no such distinctive
feature. We observe the doubly-peaked local rapidity distribution.
