Vortex singularities in acoustic and electromagnetic fields are instrumental degrees of freedom in advanced wavefront-shaping schemes and robust high-throughput communications. Laser sources that emit coherent vortices in free space have been demonstrated at optical frequencies, however their microwave counterparts, vortex masers , have remained entirely unexplored, despite their promising application potential as low-noise quantum sources and sensors. Here, we demonstrate a room-temperature maser emitting pulses of electromagnetic radiation with polarization and phase vortices, based on the physics of 3D topological vectorial singularities. Nontrivial microwave photons with polarization winding are emitted from a maser made of a subwavelength dielectric cavity filled with an organic gain medium. By topping the cavity with a chiral metasurface, the circular polarizations decouple, allowing the masing of pulses with nonzero orbital angular momentum revealed through nontrivial wavefront winding. Our work paves the way for multidimensional vortex and singularity emission from volumetric coherent microwave sources, topological photonic radiation, and novel practical applications of masers.