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 nonequilibrium 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.