Using the wide multiband photometry available in the Cosmic Evolution Survey (COSMOS) field, we explore the host galaxy properties of a large sample of active galactic nuclei (AGNs; ∼1700 objects) with Lbol ranging from 1043 to 1047 erg s−1, obtained by combining X-ray and optical spectroscopic selections. Based on a careful study of their spectral energy distributions, which have been parametrized using a two-component (AGN+galaxy) model fit, we have derived dust-corrected rest-frame magnitudes, colours and stellar masses of the obscured and unobscured AGN hosts up to high redshift (). Moreover, for the sample of obscured AGNs, we have also derived reliable star formation rates (SFRs). We find that AGN hosts span a large range of stellar masses and SFRs. No colour-bimodality is seen at any redshift in the AGN hosts, which are found to be mainly massive, red galaxies. Once we have accounted for the colour–mass degeneracy in well-defined mass-matched samples, we find a residual (marginal) enhancement of the incidence of AGNs in redder galaxies with lower specific SFRs. We argue that this result might emerge because of our ability to properly account for AGN light contamination and dust extinction, compared to surveys with a more limited multiwavelength coverage. However, because these colour shifts are relatively small, systematic effects could still be considered responsible for some of the observed trends. Interestingly, we find that the probability for a galaxy to host a black hole that is growing at any given ‘specific accretion rate’ (i.e. the ratio of X-ray luminosity to the host stellar mass) is almost independent of the host galaxy mass, while it decreases as a power law with LX/M*. By analysing the normalization of such a probability distribution, we show how the incidence of AGNs increases with redshift as rapidly as (1 + z)4, which closely resembles the overall evolution of the specific SFR of the entire galaxy population. We provide analytical fitting formulae that describe the probability of a galaxy of any mass (above the completeness limit of the COSMOS) to host an AGN of any given specific accretion rate as a function of redshift. These can be useful tools for theoretical studies of the growing population of black holes within galaxy evolution models. Although AGN activity and star formation in galaxies do appear to have a common triggering mechanism, at least in a statistical sense, within the COSMOS sample, we do not find any conclusive evidence to suggest that AGNs have a powerful influence on the star-forming properties of their host galaxies.