Mass assembly and morphological transformations since $z\sim3$ from CANDELS

[abridged] We quantify the evolution of the stellar mass functions of star-forming and quiescent galaxies as a function of morphology from $z\sim 3$ to the present. Our sample consists of ~50,000 galaxies in the CANDELS fields ($\sim880$ $arcmin^2$), which we divide into four main morphological types, i.e. pure bulge dominated systems, pure spiral disk dominated, intermediate 2-component bulge+disk systems and irregular disturbed galaxies. Our main results are: Star-formation: At $z\sim 2$, 80\% of the stellar mass density of star-forming galaxies is in irregular systems. However, by $z\sim 0.5$, irregular objects only dominate at stellar masses below $10^9M\odot$. A majority of the star-forming irregulars present at $z\sim 2$ undergo a gradual transformation from disturbed to normal spiral disk morphologies by $z\sim 1$ 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. Quenching: We confirm that galaxies reaching a stellar mass of $M_*\sim10^{10.8}M_\odot$ ($M^*$) tend to quench. Also, quenching implies the presence of a bulge: the abundance of massive red disks 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 above $M^*$ is dominated by compact spheroids. Quenching at this early epoch destroys the disk 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 disks with a significant central bulge. This suggests that mass-quenching at this epoch starts from the inner parts and preserves the disk. At $z<1$, the high mass end of the passive SMF is globally in place and the evolution mostly happens at stellar masses below $10^{10}M_\odot$.