Cadenced optical imaging surveys in the next decade will be capable of detecting time-varying galaxy-scale strong gravitational lenses in large numbers, increasing the size of the statistically well-defined samples of multiply imaged quasars by two orders of magnitude, and discovering the first strongly lensed supernovae. We carry out a detailed calculation of the likely yields of several planned surveys, using realistic distributions for the lens and source properties and taking magnification bias and image configuration detectability into account. We find that upcoming wide-field synoptic surveys should detect several thousand lensed quasars. In particular, the Large Synoptic Survey Telescope (LSST) should find more than some 8000 lensed quasars, some 3000 of which will have well-measured time delays. The LSST should also find some 130 lensed supernovae during the 10-yr survey duration, which is compared with ∼15 lensed supernovae predicted to be found by a deep, space-based supernova survey done by the Joint Dark Energy Mission. We compute the quad fraction in each survey, predicting it to be ∼15 per cent for the lensed quasars and ∼30 per cent for the lensed supernovae. Generating a mock catalogue of around 1500 well-observed double-image lenses, as could be derived from the LSST survey, we compute the available precision on the Hubble constant and the dark energy equation parameters for the time-delay distance experiment (assuming priors from Planck): the predicted marginalized 68 per cent confidence intervals are σ(w0) = 0.15, σ(wa) = 0.41 and σ(h) = 0.017. While this is encouraging in the sense that these uncertainties are only 50 per cent larger than those predicted for a space-based Type Ia supernova sample, we show how the dark energy figure of merit degrades with decreasing knowledge of the lens mass distribution.