Microcavity polaritons in the lasing regime undergo a spontaneous symmetry-breaking transition resulting in coherent emission with a well-defined polarization. The order parameter is thus a vector describing both the laser global phase and its polarization. Using an ultrafast single-shot detection technique, we show that polariton lasing in GaAs-based microcavities presents a high degree of second-order coherence [g(2)(τ=0)≈1] above threshold, and that the initial polarization is stochastic, taking any possible direction in the Poincaré sphere (linear, elliptical, or circular). Once the polarization direction is established, subsequent oscillations of the emission probability witness the presence of an intrinsic polarization splitting. These results show the intricate polarization dynamics in the onset of polariton lasers.