We present a combined X-ray and optical analysis of three bimodal galaxy clusters selected as merging candidates at z similar to 0.1. These targets are part of MUSIC (MUlti-Wavelength Sample of Interacting Clusters), which is a general project designed to study the physics of merging clusters by means of multi-wavelength observations. Observations include spectro-imaging with XMM-Newton EPIC camera, multi-object spectroscopy (260 new redshifts), and wide-field imaging at the ESO 3.6 m and 2.2 m telescopes. We build a global picture of these clusters using X-ray luminosity and temperature maps together with galaxy density and velocity distributions. Idealized numerical simulations were used to constrain the merging scenario for each system. We show that A2933 is very likely an equal-mass advanced pre-merger similar to 200 Myr before the core collapse, while A2440 and A2384 are post-merger systems (similar to 450 Myr and similar to 1.5 Gyr after core collapse, respectively). In the case of A2384, we detect a spectacular filament of galaxies and gas spreading over more than 1 h(-1) Mpc, which we infer to have been stripped during the previous collision. The analysis of the MUSIC sample allows us to outline some general properties of merging clusters: a strong luminosity segregation of galaxies in recent post-mergers; the existence of preferential axes - corresponding to the merging directions - along which the BCGs and structures on various scales are aligned; the concomitance, in most major merger cases, of secondary merging or accretion events, with groups infalling onto the main cluster, and in some cases the evidence of previous merging episodes in one of the main components. These results are in good agreement with the hierarchical scenario of structure formation, in which clusters are expected to form by successive merging events, and matter is accreted along large-scale filaments.