ABSTRACT We perform a structural decomposition of galaxies identified in three cosmological hydrodynamical simulations by applying Gaussian mixture models (GMMs) to the kinematics of their stellar particles. We study the resulting disc, bulge, and intra-halo light (IHL) components of galaxies whose host dark matter haloes have virial masses in the range M200 = 1011–$10^{15}\, {\rm M_⊙ }$. Our decomposition technique isolates galactic discs whose mass fractions, fdisc, correlate strongly with common alternative morphology indicators; for example, fdisc is approximately equal to κco, the fraction of stellar kinetic energy in corotation. The primary aim of our study, however, is to characterize the IHL of galaxies in a consistent manner and over a broad mass range, and to analyse its properties from the scale of galactic stellar haloes up to the intra-cluster light. Our results imply that the IHL fraction, fIHL, has appreciable scatter and is strongly correlated with galaxy morphology: at fixed stellar mass, the IHL of disc galaxies is typically older and less massive than that of spheroids. Above $M_{200}≈ 10^{13}\, {\rm M_⊙ }$, we find, on average, fIHL ≈ 0.37, albeit with considerable scatter. The transition radius beyond which the IHL dominates the stellar mass of a galaxy is roughly $30\, {\rm kpc}$ for disc galaxies, but depends strongly on halo mass for spheroids. However, we find that no alternative IHL definitions – whether based on the ex situ stellar mass, or the stellar mass outside a spherical aperture – reproduce our dynamically defined IHL masses.