  We study, theoretically and numerically, the onset and development of
modulational instability in an incoherently pumped spatially homogeneous
polariton condensate. Within the framework of mean-field theory, we identify
regimes of modulational instability in two cases: 1) Strong feedback between
the condensate and reservoir, which may occur in scalar condensates, and 2)
Parametric scattering in the presence of polarization splitting in spinor
condensates. In both cases we investigate the instability induced textures in
space and time including non-equilibrium dynamics of phase dislocations and
vortices. In particular we discuss the mechanism of vortex destabilization and
formation of spiraling waves. We also identify the presence of topological
defects, which take the form of half-vortex pairs in the spinor case, giving an
"eyelet" structure in intensity and dipole type structure in the spin
polarization. In the modulationally stable parameter domains, we observe
formation of the phase defects in the process of condensate formation from an
initially spatially incoherent low-density state. In analogy to the
Kibble-Zurek type scaling for nonequilibrium phase transitions, we find that
the defect density scales with the pumping rate.
