  We present estimates for the number of shadow-resolved supermassive black
hole (SMBH) systems that can be detected using radio interferometers, as a
function of angular resolution, flux density sensitivity, and observing
frequency. Accounting for the distribution of SMBHs across mass, redshift, and
accretion rate, we use a new semi-analytic spectral energy distribution model
to derive the number of SMBHs with detectable and optically thin horizon-scale
emission. We demonstrate that (sub)millimeter interferometric observations with
${\sim}0.1$ $\mu$as resolution and ${\sim}1$ $\mu$Jy sensitivity could access
${>}10^6$ SMBH shadows. We then further decompose the shadow source counts into
the number of black holes for which we could expect to observe the first- and
second-order lensed photon rings. Accessing the bulk population of first-order
photon rings requires ${\lesssim}2$ $\mu$as resolution and ${\lesssim}0.5$ mJy
sensitivity, while doing the same for second-order photon rings requires
${\lesssim}0.1$ $\mu$as resolution and ${\lesssim}5$ $\mu$Jy sensitivity. Our
model predicts that with modest improvements to sensitivity, as many as $\sim$5
additional horizon-resolved sources should become accessible to the current
Event Horizon Telescope (EHT), while a next-generation EHT observing at 345 GHz
should have access to ${\sim}$3 times as many sources. More generally, our
results can help guide enhancements of current arrays and specifications for
future interferometric experiments that aim to spatially resolve a large
population of SMBH shadows or higher-order photon rings.
