From cosmology to the microscopic scales of the quantum world, the study of topological excitations is essential for the understanding of phase conformation and phase transitions. Quantum fluids are convenient systems to investigate topological entities because well-established techniques are available for their preparation, control and measurement. Across a phase transition, a system dramatically changes its properties because of the spontaneous breaking of certain continuous symmetries, leading to generation of topological defects. In particular, attention is given to entities that involve both spin and phase topologies. Exciton-polariton condensates are quantum fluids combining coherence and spin properties that, thanks to their light-matter nature, bring the advantage of direct optical access to the condensate order parameter. Here we report on the spontaneous occurrence of hyperbolic spin vortices in polariton condensates, by directly imaging both their phase and spin structure, and observe the associated spatial polarization patterns, spin textures that arise in the condensate.