Supernovae that are strongly gravitationally lensed (gLSNe) by galaxies are powerful probes of astrophysics and cosmology that will be discovered systematically by next-generation wide-field, high-cadence imaging surveys such as the Zwicky Transient Facility (ZTF) and the Large Synoptic Survey Telescope (LSST). Here we use pixel-level simulations that include dust, observing strategy, and multiple supernova subtypes to forecast the rates and properties of gLSNe that ZTF and LSST will find. Applying the resolution-insensitive discovery strategy of Goldstein et al. (2018), we forecast that ZTF (LSST) can discover 0.02 (0.79) 91bg-like, 0.17 (5.92) 91T-like, 1.22 (47.84) Type Ia, 2.76 (88.51) Type IIP, 0.31 (12.78) Type IIL, and 0.36 (15.43) Type Ib/c gLSNe per year. We also forecast that the surveys can discover at least 3.75 (209.32) Type IIn gLSNe per year, for a total of at least 8.60 (380.60) \glsne\ per year under fiducial observing strategies. ZTF gLSNe have a median $z_s=0.9$, $z_l=0.35$, $\mu_\mathrm{tot}=30$, $\Delta t_\mathrm{max}= 10$ days, $\min(\theta)= 0.25^{\prime\prime}$, and $N_\mathrm{img} = 4$. LSST gLSNe are less compact and less magnified, with a median $z_s=1.0$, $z_l=0.4$, $\mu_\mathrm{tot}\approx6$, $\Delta t_\mathrm{max} = 25$ days, $\min(\theta)=0.6^{\prime\prime}$, and $N_\mathrm{img} = 2$. As the properties of lensed host galaxy arcs provide critical information for lens mass modeling, we develop a model of the supernova--host galaxy connection and use it to simulate realistic images of the supernova--host--lens systems. We find that the vast majority of gLSN host galaxies will be multiply imaged, enabling detailed constraints on lens models with sufficiently deep high-resolution imaging taken after the supernova has faded. We release the results of our simulations to the public as catalogs at this URL: this http URL.
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