We show that when unilamellar polymer vesicles dispersed in water made from a blockcopolymer, in this case poly((ethylene oxide)45-block-(methyl methacrylate)164), poly((ethylene oxide)45-block-(methyl methacrylate)170), or poly(n-butyl methacrylate)81-block-(2-(dimethylamino)ethyl methacrylate)20, are exposed to small hydrophobic molecules, here methyl methacrylate as well as n-butyl methacrylate, they can undergo morphological transitions. Upon swelling, the polymersomes lose their original simple bilayer morphology and transform into more complex coil-like and patchy colloidal structures, as investigated experimentally by cryogenic electron microscopy (cryo-EM). Dissipative particle dynamics (DPD) simulations on a model flat bilayer indeed show that transitions can occur upon bilayer swelling, which is accompanied by a change in the mechanical bilayer properties. The transition involves the formation of water pockets in the interior regions of the bilayer. Co-existence of the various morphologies in the experiments suggests an activation barrier towards morphological changes and a possibility of multiple meta-stable states. The latter indeed is supported by the existence of multiple minima in the surface tension as a function of bilayer area, as found in the simulations.