We examine the spheroid growth and star formation quenching experienced by galaxies since z ∼ 3 by studying the evolution with redshift of the quiescent and spheroid-dominated fractions of galaxies from the CANDELS (Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey) and GAMA (Galaxy and Mass Assembly) surveys. We compare the observed fractions with predictions from a semi-analytic model which includes prescriptions for bulge growth and AGN feedback due to mergers and disc instabilities. We facilitate direct morphological comparison by converting our model bulge-to-total stellar mass ratios to Sérsic indices. We then subdivide our population into the four quadrants of the specific star formation rate–Sérsic index plane and study the build-up of each of these subpopulations. We find that the fraction of star-forming discs declines steadily, while the fraction of quiescent spheroids builds up over cosmic time. The fractions of star-forming spheroids and quiescent discs are both non-negligible, and stay nearly constant over the period we have studied. Our model is qualitatively successful at reproducing the evolution of the two ‘main’ populations (star-forming discs and quiescent spheroids), and approximately reproduces the relative fractions of all four types, but predicts a stronger decline in star-forming spheroids, and increase in quiescent discs, than is seen in the observations. A model with an additional channel for bulge growth via disc instabilities agrees better overall with the observations than a model in which bulges can grow only through mergers. We also examine the relative importance of these different physical drivers of transformation (major and minor mergers and disc instabilities).