Context. The interaction between stellar winds and the interstellar medium (ISM) can create complex bow shocks. The photometers on board the Herschel Space Observatory are ideally suited to studying the morphologies of these bow shocks. Aims. We aim to study the circumstellar environment and wind-ISM interaction of the nearest red supergiant, Betelgeuse. Methods. Herschel PACS images at 70, 100, and 160 μm and SPIRE images at 250, 350, and 500 μm were obtained by scanning the region around Betelgeuse. These data were complemented with ultraviolet GALEX data, near-infrared WISE data, and radio 21 cm GALFA-HI data. The observational properties of the bow shock structure were deduced from the data and compared with hydrodynamical simulations. Results. The infrared Herschel images of the environment around Betelgeuse are spectacular, showing the occurrence of multiple arcs at ∼6-7 from the central target and the presence of a linear bar at ∼9 . Remarkably, no large-scale instabilities are seen in the outer arcs and linear bar. The dust temperature in the outer arcs varies between 40 and 140 K, with the linear bar having the same colour temperature as the arcs. The inner envelope shows clear evidence of a non-homogeneous clumpy structure (beyond 15 ), probably related to the giant convection cells of the outer atmosphere. The non-homogeneous distribution of the material even persists until the collision with the ISM. A strong variation in brightness of the inner clumps at a radius of ∼2 suggests a drastic change in mean gas and dust density ∼32 000 yr ago. Using hydrodynamical simulations, we try to explain the observed morphology of the bow shock around Betelgeuse. Conclusions. Different hypotheses, based on observational and theoretical constraints, are formulated to explain the origin of the multiple arcs and the linear bar and the fact that no large-scale instabilities are visible in the bow shock region. We infer that the two main ingredients for explaining these phenomena are a non-homogeneous mass-loss process and the influence of the Galactic magnetic field. The hydrodynamical simulations show that a warm interstellar medium, reflecting a warm neutral or partially ionized medium, or a higher temperature in the shocked wind also prevent the growth of strong instabilities. The linear bar is probably an interstellar structure illuminated by Betelgeuse itself.