  The objects that emit extragalatic fast radio bursts (FRBs) remain
unidentified. Studies of the host galaxies and environments of accurately
localised ($\lesssim1$ 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 $\sim0.3$ dex, and
the median $M_{*}$ by $\sim0.5$ dex. The median projected physical offset is
increased by $\sim2$ kpc ($\sim0.25$ 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.
