We report an experimental test of single-shot polarimetry applied to the problem of real-time monitoring of the output polarization states in each core within a multicore fiber bundle. The technique uses a stress-engineered optical element, together with an analyzer, and provides a point spread function whose shape unambiguously reveals the polarization state of a point source. We implement this technique to monitor, simultaneously and in real time, the output polarization states of up to 180 single-mode fiber cores in both conventional and polarization-maintaining fiber bundles. We demonstrate also that the technique can be used to fully characterize the polarization properties of each individual fiber core, including eigen-polarization states, phase delay, and diattenuation.