  Spinorial or multi-component Bose-Einstein condensates may sustain fractional
quanta of circulation, vorticant topological excitations with half integer
windings of phase and polarization. Matter-light quantum fluids, such as
microcavity polaritons, represent a unique test bed for realising strongly
interacting and out-of-equilibrium condensates. The direct access to the phase
of their wavefunction enables us to pursue the quest of whether half vortices
---rather than full integer vortices--- are the fundamental topological
excitations of a spinor polariton fluid. Here, we are able to directly generate
by resonant pulsed excitations, a polariton fluid carrying either the half or
full vortex states as initial condition, and to follow their coherent evolution
using ultrafast holography. Surprisingly we observe a rich phenomenology that
shows a stable evolution of a phase singularity in a single component as well
as in the full vortex state, spiraling, splitting and branching of the initial
cores under different regimes and the proliferation of many vortex anti-vortex
pairs in self generated circular ripples. This allows us to devise the
interplay of nonlinearity and sample disorder in shaping the fluid and driving
the phase singularities dynamics
