  The Legacy Survey of Space and Time (LSST) by the Vera C. Rubin Observatory
is expected to discover tens of millions of quasars. A significant fraction of
these could be powered by coalescing massive black hole (MBH) binaries, since
many quasars are believed to be triggered by mergers. We show that under
plausible assumptions about the luminosity functions, lifetimes, and binary
fractions of quasars, we expect the full LSST quasar catalogue to contain
between 20-100 million compact MBH binaries with masses $M=10^{5-9}M_{\odot}$,
redshifts $z=0-6$, and orbital periods $P=1-70$ days. Their light-curves are
expected to be distinctly periodic, which can be confidently distinguished from
stochastic red-noise variability, because LSST will cover dozens, or even
hundreds of cycles. A very small subset of 10-150 ultra-compact ($P\lesssim1$
day) binary quasars among these will, over $\sim$5-15 years, evolve into the
mHz gravitational-wave (GW) frequency band and can be detected by
$\textit{LISA}$. They can therefore be regarded as "$\textit{LISA}$
verification binaries", analogous to short-period Galactic compact-object
binaries. The practical question is how to find these handful of "needles in
the haystack" among the large number of quasars: this will likely require a
tailored co-adding analysis optimised for this purpose.
