The analysis of galaxies on the star formation rate-stellar mass (SFR–M∗) plane is a powerful diagnostic for galaxy evolution at different cosmic times. We consider a sample of 24 463 galaxies from the CANDELS/GOODS-S survey to conduct a detailed analysis of the SFR–M∗ relation at redshifts \hbox{$0.5\leqslant z<$}0.5 ⩽ z<3 over more than three dex in stellar mass. To obtain SFR estimates, we utilise mid- and far-IR photometry when available, and rest-UV fluxes for all the other galaxies. We perform our analysis in different redshift bins, with two different methods: 1) a linear regression fitting of all star-forming galaxies, defined as those with specific SFRs log 10(sSFR/ yr-1) > −9.8, similarly to what is typically done in the literature; 2) a multi-Gaussian decomposition to identify the galaxy main sequence (MS), the starburst sequence and the quenched galaxy cloud. We find that the MS slope becomes flatter when higher stellar mass cuts are adopted, and that the apparent slope change observed at high masses depends on the SFR estimation method. In addition, the multi-Gaussian decomposition reveals the presence of a starburst population which increases towards low stellar masses and high redshifts. We find that starbursts make up ~ 5% of all galaxies at z = 0.5−1.0, while they account for ~ 16% of galaxies at 2 <z< 3 with log10(M∗/M0) = 8.25–11.25. We conclude that the dissection of the SFR–M∗ in multiple components over a wide range of stellar masses is necessary to understand the importance of the different modes of star formation through cosmic time.