  The realization that first- and second-generation Yukawa couplings can be
probed by decays of the Higgs boson to a meson in association with a photon has
renewed interest in such rare exclusive decays. We present here a detailed
study of the rare $Z$-boson processes $Z \to J/\psi+\gamma$, $Z \to
\Upsilon+\gamma$, and $Z \to \phi+\gamma$ that can serve as benchmarks for the
analogous Higgs-boson decays. We include both direct-production and
fragmentation contributions to these decays, and consider the leading QCD
corrections and the relativistic corrections to the $J/\psi$ and $\Upsilon$
processes. We present numerical predictions for the branching ratios that
include a careful accounting of the theoretical uncertainties.
