We analyze solutions drawn from the recently published posterior distribution of the TRAPPIST-1 system, which consists of seven Earth-size planets appearing to be in a resonant chain around a red dwarf. We show that all the planets are simultaneously in two-planet and three-planet resonances, apart from the innermost pair for which the two-planet resonant angles circulate. By means of a frequency analysis, we highlight that the transit-timing variation (TTV) signals possess a series of common periods varying from days to decades, which are also present in the variations of the dynamical variables of the system. Shorter periods (e.g., the TTVs characteristic timescale of 1.3 yr) are associated with two-planet mean-motion resonances, while longer periods arise from three-planet resonances. By use of N-body simulations with migration forces, we explore the origin of the resonant chain of TRAPPIST-1 and find that for particular disc conditions, a chain of resonances - similar to the observed one - can be formed which accurately reproduces the observed TTVs. Our analysis suggests that while the 4-yr collected data of observations hold key information on the two-planet resonant dynamics, further monitoring of TRAPPIST-1 will soon provide signatures of three-body resonances, in particular the 3.3 and 5.1 yr periodicities expected for the current best-fit solution. Additional observations would help to assess whether the innermost pair of planets is indeed resonant (its proximity to the 8:5 resonance being challenging to explain), and therefore give additional constraints on formation scenarios.