ABSTRACT Fast radio bursts (FRBs) are the first cosmological radio sources that vary on millisecond time-scales, 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, 20–40 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 63–85 per cent of ASKAP FRBs found in its coherent search mode. CHIME FRBs will reach z ∼ 3, SKA1-Mid FRBs z ∼ 5, but ground based follow-up is limited to z ≲ 1.5. We discuss the consequences for several FRB applications. If ∼1/2 of ASKAP FRBs have measured redshifts, 1000 detected FRBs can be used to constrain Ωbh70 to within ∼10 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.