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We quantify the evolution of the stellar mass functions (SMFs) of star-forming and quiescent galaxies as a function of morphology from z ∼ 3 to the present. Our sample consists of ∼50 000 galaxies in the CANDELS fields (∼880 arcmin2), which we divide into four main morphological types, i.e. pure bulge-dominated systems, pure spiral disc-dominated, intermediate two-component bulge+disc systems and irregular disturbed galaxies. At z ∼ 2, 80 per cent of the stellarmass density of star-forming galaxies is in irregular systems. However, by z ∼ 0.5, irregular objects only dominate at stellar masses below 109M_. A majority of the star-forming irregulars present at z ∼ 2 undergo a gradual transformation from disturbed to normal spiral disc morphologies by z ∼ 1 without significant interruption to their star formation. Rejuvenation after a quenching event does not seem to be common except perhaps for the most massive objects, because the fraction of bulge-dominated star-forming galaxies with M∗/M_ > 1010.7 reaches 40 per cent at z < 1. Quenching implies the presence of a bulge: the abundance of massive red discs is negligible at all redshifts over 2 dex in stellar mass. However, the dominant quenching mechanism evolves. At z > 2, the SMF of quiescent galaxies aboveM ∗ is dominated by compact spheroids. Quenching at this early epoch destroys the disc and produces a compact remnant unless the star-forming progenitors at even higher redshifts are significantly more dense. At 1 < z<2, the majority of newly quenched galaxies are discs with a significant central bulge. This suggests that mass quenching at this epoch starts from the inner parts and preserves the disc. At z < 1, the high-mass end of the passive SMF is globally in place and the evolution mostly happens at stellar masses below 1010M_. These low-mass galaxies are compact, bulge-dominated systems, which were environmentally quenched: destruction of the disc through ram-pressure stripping is the likely process.