We investigate the intriguing phenomenon of beta decay of a free neutron in a non-plane-wave (structured) state. Our analysis covers three types of states: unpolarized vortex (Bessel) neutrons that possess nonzero orbital angular momentum (OAM), Laguerre-Gaussian wave packets, and spin-correlated OAM (spin-orbit) states characterized by unique polarization patterns. These states are of particular interest as they have recently been generated in neutron optics experiments and have promising applications in studies of quantum magnetic materials. The spectral-angular distributions (SAD) of the emitted electrons and protons are examined. We show that the high sensitivity of the protons' SAD to the structure of the neutron wave packet can be used as a tool to extract the distinctive features of the non-plane-wave neutron states. Furthermore, we demonstrate that the angular distribution of the emitted particles serves as a reflection of the spatial symmetries inherent to the neutron wave packet.