  Fast radio bursts (FRBs) are the first cosmological radio sources that vary
on millisecond timescales, which makes them a unique probe of the Universe.
Many proposed applications of FRBs require associated redshifts. These can only
be obtained by localizing FRBs to their host galaxies and subsequently
measuring their redshifts. Upcoming FRB surveys will provide arcsecond
localization for many FRBs, not all of which can be followed up with dedicated
optical observations. We aim to estimate the fraction of FRB hosts that will be
catalogued with redshifts by existing and future optical surveys. We use the
population synthesis code frbpoppy to simulate several FRB surveys, and the
semi-analytical galaxy formation code GALFORM to simulate their host galaxies.
We obtain redshift distributions for the simulated FRBs and the fraction with
host galaxies in a survey. Depending on whether FRBs follow the cosmic star
formation rate or stellar mass, 20 to 40 per cent of CHIME FRB hosts will be
observed in an SDSS-like survey, all at $z<0.5$. The deeper DELVE survey will
detect 63 to 85 per cent of ASKAP FRBs found in its coherent search mode. CHIME
FRBs will reach $z\sim 3$, SKA1-Mid FRBs $z\sim 5$, but ground based follow-up
is limited to $z\lesssim 1.5$. We discuss consequences for several FRB
applications. If $\sim1/2$ of ASKAP FRBs have measured redshifts, 1000 detected
FRBs can be used to constrain $\Omega_\text{b} h_{70}$ to within $\sim10$ per
cent at 95 per cent credibility. We provide strategies for optimized follow-up,
when building on data from existing surveys. Data and codes are made available.
