Using data from four deep fields (COSMOS, AEGIS, ECDFS, and CDFN), we study the correlation between the position of galaxies in the star formation rate (SFR) versus stellar mass plane and local environment at z < 1.1. To accurately estimate the galaxy SFR, we use the deepest available Spitzer/MIPS 24 and Herschel/PACS data sets. We distinguish group environments (...) based on the available deep X-ray data and lower halo mass environments based on the local galaxy density. We confirm that the main sequence (MS) of star-forming galaxies is not a linear relation and there is a flattening towards higher stellar masses (...), across all environments. At high redshift (0.5 < z < 1.1), the MS varies little with environment. At low redshift (0.15 < z < 0.5), group galaxies tend to deviate from the mean MS towards the region of quiescence with respect to isolated galaxies and less-dense environments. We find that the flattening of the MS towards low SFR is due to an increased fraction of bulge-dominated galaxies at high masses. Instead, the deviation of group galaxies from the MS at low redshift is caused by a large fraction of red disc-dominated galaxies which are not present in the lower density environments. Our results suggest that above a mass threshold (...) stellar mass, morphology and environment act together in driving the evolution of the star formation activity towards lower level. The presence of a dominating bulge and the associated quenching processes are already in place beyond z ~ 1. The environmental effects appear, instead, at lower redshifts and have a long time-scale. (ProQuest: ... denotes formulae/symbols omitted.)