  In this paper we discuss, within the Gross--Pitaevskii framework,
superfluidity, soliton nucleation, and instabilities in a non-equilibrium
polariton fluid injected by a spatially localized and continuous-wave coherent
pump and flowing against a defect located outside the pump spot. In contrast to
equilibrium condensates, the steady-state solutions of the driven-dissipative
equations in this specific geometry hardly show a clean superfluid flow around
the defect and rather feature a crossover from shallow to deep soliton-like
perturbation. This is explained in terms of the properties of one-dimensional
flows, in particular their weak dependence on the pump parameters and their
rapid transition to a super-sonic regime under the effect of the quantum
pressure; such a highly nonlinear behaviour calls for quantitative experimental
tests of the underlying Gross--Pitaevskii equation. The role of disorder and of
a incoherent reservoir in inducing non-stationary behaviours with moving
vortices is also highlighted.
