ABSTRACT We present a complete structural analysis of the ellipticals (E), diffuse bulges (dB), compact bulges (cB), and discs (D) within a redshift range 0 < z < 1, and stellar mass log10(M*/M⊙) ≥ 9.5 volume-limited sample drawn from the combined DEVILS and HST-COSMOS region. We use the profit code to profile over ∼35 000 galaxies for which visual classification into single or double component was pre-defined in Paper-I. Over this redshift range, we see a growth in the total stellar mass density (SMD) of a factor of 1.5. At all epochs we find that the dominant structure, contributing to the total SMD, is the disc, and holds a fairly constant share of $∼ 60{{\ \rm per\ cent}}$ of the total SMD from z = 0.8 to z = 0.2, dropping to $∼ 30{{\ \rm per\ cent}}$ at z = 0.0 (representing $∼ 33{{\ \rm per\ cent}}$ decline in the total disc SMD). Other classes (E, dB, and cB) show steady growth in their numbers and integrated stellar mass densities. By number, the most dramatic change across the full mass range is in the growth of diffuse bulges. In terms of total SMD, the biggest gain is an increase in massive elliptical systems, rising from 20 per cent at z = 0.8 to equal that of discs at z = 0.0 (30 per cent) representing an absolute mass growth of a factor of 2.5. Overall, we see a clear picture of the emergence and growth of all three classes of spheroids over the past 8 Gyr, and infer that in the later half of the Universe’s timeline spheroid-forming processes and pathways (secular evolution, mass-accretion, and mergers) appear to dominate mass transformation over quiescent disc growth.