The relation between the stellar mass (M⋆) and the star formation rate (SFR) characterizes how the instantaneous star formation is determined by the galaxy past star formation history and by the growth of the dark matter structures. We deconstruct the M⋆-SFR plane by measuring the specific SFR functions in several stellar mass bins from z = 0.2 out to z = 1.4 (specific SFR = SFR/M⋆, noted sSFR). Our analysis is primary based on a 24 μm selected catalogue combining the COSMOS and GOODS surveys. We estimate the SFR by combining mid- and far-infrared data for 20500 galaxies. The sSFR functions are derived in four stellar mass bins within the range 9.5 < log (M⋆ /M⊙) < 11.5. First, we demonstrate the importance of taking into account selection effects when studying the M⋆-SFR relation. Secondly, we find a mass-dependent evolution of the median sSFR with redshift varying as sSFR ∝ (1 + z)^b, with b increasing from b = 2.88^(±0.12) to b = 3.78^(± 0.60) between M⋆ = 10^(9.75) M⊙ and M⋆ = 10^(11.1) M⊙, respectively. At low masses, this evolution is consistent with the cosmological accretion rate and predictions from semi-analytical models (SAM). This agreement breaks down for more massive galaxies showing the need for a more comprehensive description of the star formation history in massive galaxies. Third, we obtain that the shape of the sSFR function is invariant with time at z< 1.4 but depends on the mass. We observe a broadening of the sSFR function ranging from 0.28 dex at M⋆ = 10^(9.75) M⊙ to 0.46 dex at M⋆ = 10^(11.1) M⊙. Such increase in the intrinsic scatter of the M⋆-SFR relation suggests an increasing diversity ofstar formation histories (SFHs) as the stellar mass increases. Finally, we find a gradual decline of the sSFR with stellar mass as log _(10)(sSFR) ∝ −0.17M⋆. We discuss the numerous physical processes, as gas exhaustion in hot gas halos or secular evolution, which can gradually reduce the sSFR and increase the SFH diversity.