The objects that emit extragalatic fast radio bursts (FRBs) remain unidentified. Studies of the host galaxies and environments of accurately localised ( 1 arcsec) FRBs promise to deliver critical insights into the nature of their progenitors. Here we demonstrate the effects of observational selection biases on analyses of the distributions of FRB host-galaxy properties (including star-formation rate, SFR, and stellar mass, 𝑀 * ), and on the distributions of FRB offsets from the centres of their hosts. We consider the effects of "radio selection", wherein FRBs with larger dispersion measures and scattering timescales are less likely to be detected, and the effects of "optical selection", wherein FRBs with fainter host galaxies are more likely to have unidentified or mis-identified hosts. We develop a plausible, illustrative model for these effects in observations of FRBs and their host galaxies by combining the output catalogues of a semi-analytic galaxy formation model with a recently developed algorithm to associate FRBs with host galaxies (PATH). We find that optical selection biases are most important for the host-galaxy 𝑀 * and SFR distributions, and that radio selection biases are most important for the distribution of FRB projected physical offsets. For our fiducial simulation of FRBs at 𝑧 < 0.5, the selection biases cause the median host-galaxy SFR to be increased by ∼ 0.3 dex, and the median 𝑀 * by ∼ 0.5 dex. The median projected physical offset is increased by ∼ 2 kpc (∼ 0.25 dex). These effects are sufficiently large so as to merit careful consideration in studies of localised FRBs, and our simulations provide a guide towards their mitigation.