ABSTRACT Compact groups (CGs) of galaxies are defined as isolated and dense galaxy systems that appear to be a unique site of multiple galaxy interactions. Semi-analytical models (SAMs) of galaxy formation are a prime tool to understand CGs. We investigate how the frequency and the 3D nature of CGs depends on the SAM and its underlying cosmological parameters. Extracting nine light-cones of galaxies from five different SAMs and selecting CGs as in observed samples, we find that the frequency and nature of CGs depends strongly on the cosmological parameters. Moving from the WMAP1 to the WMAP7 and Planck cosmologies (increasing density of the Universe and decreasing normalization of the power spectrum), the space density of CGs is decreased by a factor 2.5, while the fraction of CGs that are physically dense falls from 50 to 35 per cent. The lower σ8 leads to fewer dense groups, while the higher Ωm causes more chance alignments. However, with increased mass and spatial resolution, the fraction of CGs that are physically dense is pushed back up to 50 per cent. The intrinsic differences in the SAM recipes also lead to differences in the frequency and nature of CGs, particularly those related to how SAMs treat orphan galaxies. We find no dependence of CG properties on the flux limit of the mock catalogues nor on the waveband in which galaxies are selected. One should thus be cautious when interpreting a particular SAM for the frequency and nature of CGs.