We provide a coherent, uniform measurement of the evolution of the logarithmic star formation rate (SFR) - stellar mass ($M_*$) relation, called the main sequence of star-forming galaxies (MS), for galaxies out to $zsim5$. We measure the MS using mean stacks of 3 GHz radio continuum images to derive average SFRs for $sim$200,000 mass-selected galaxies at $z>0.3$ in the COSMOS field. We describe the MS relation adopting a new model that incorporates a linear relation at low stellar mass (log($M_*$/M$_odot$)$<$10) and a flattening at high stellar mass that becomes more prominent at low redshift ($z<1.5$). We find that the SFR density peaks at $1.5<z<2$ and at each epoch there is a characteristic stellar mass ($M_* = 1 - 4 times 10^{10}mathrm{M}_odot$) that contributes the most to the overall SFR density. This characteristic mass increases with redshift, at least to $zsim2.5$. We find no significant evidence for variations in the MS relation for galaxies in different environments traced by the galaxy number density at $0.3<z<3$, nor for galaxies in X-ray groups at $zsim0.75$. We confirm that massive bulge-dominated galaxies have lower SFRs than disk-dominated galaxies at a fixed stellar mass at $z<1.2$. As a consequence, the increase in bulge-dominated galaxies in the local star-forming population leads to a flattening of the MS at high stellar masses. This indicates that mass-quenching is linked with changes in the morphological composition of galaxies at a fixed stellar mass.
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