We present a series of dynamical maps for fictitious 3-planets systems in initially circular coplanar orbits. These maps have unveiled a rich resonant structure involving two or three planets, as well as indicating possible migration routes from secular to double resonances or pure 3-planet commensurabilities. These structures are then compared to the present-day orbital architecture of observed resonant chains. In a second part of the paper we describe N-body simulations of type-I migration. Depending on the orbital decay timescale, we show that 3-planet systems may be trapped in different combinations of independent commensurabilities: (i) double resonances, (ii) intersection between a 2-planet and a first-order 3-planet resonance, and (iii) simultaneous libration in two first-order 3-planet resonances. These latter outcomes are found for slow migrations, while double resonances are almost always the final outcome in high-density disks. Finally, we discuss an application to the TRAPPIST-1 system. We find that, for low migration rates and planetary masses of the order of the estimated values, most 3-planet sub-systems are able to reach the observed double resonances after following evolutionary routes defined by pure 3-planet resonances. The final orbital configuration shows resonance offsets comparable with present-day values without the need of tidal dissipation. For the 8/5 resonance proposed to dominate the dynamics of the two inner planets, we find little evidence of its dynamical significance; instead, we propose that this relation between mean motions could be a consequence of the interaction between a pure 3-planet resonance and a 2-planet commensurability between planets c and d.
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