  States of light encoding multiple polarizations - vector beams - offer unique
capabilities in metrology and communication. However, their practical
application is limited by the lack of methods for measuring many polarizations
in a scalable and compact way. Here we demonstrate polarimetry of vector beams
in a single shot without any polarization optics. We map the beam polarization
content into a spatial intensity distribution through multiple light scattering
and exploit supervised learning for single-shot measurements of multiple
polarizations. The method also allows us to classify beams with an unknown
number of polarization modes, a functionality missing in conventional
techniques. Our findings enable a fast and compact polarimeter for
polarization-structured light, a universal tool that may radically impact
optical devices for sensing, imaging, and computing.
