  We derive supernova (SN) bounds on muon-philic bosons, taking advantage of
the recent emergence of muonic SN models. Our main innovations are to consider
scalars $\phi$ in addition to pseudoscalars $a$ and to include systematically
the generic two-photon coupling $G_{\gamma\gamma}$ implied by a muon triangle
loop. This interaction allows for Primakoff scattering and radiative boson
decays. The globular-cluster bound $G_{\gamma\gamma}<0.67\times10^{-10}~{\rm
GeV}^{-1}$ derived for axion-like particles carries over to the muonic Yukawa
couplings as $g_a<3.1\times10^{-9}$ and $g_\phi< 4.6\times10^{-9}$ for
$m_{a,\phi}\lesssim 100$ keV, so SN arguments become interesting mainly for
larger masses. If bosons escape freely from the SN core the main constraints
originate from SN1987A $\gamma$ rays and the diffuse cosmic $\gamma$-ray
background. The latter allows at most $10^{-4}$ of a typical total SN energy of
$E_{\rm SN}\simeq3\times10^{53}$erg to show up as $\gamma$ rays, for
$m_{a,\phi}\gtrsim 100$keV implying $g_a \lesssim 0.9\times10^{-10}$ and
$g_\phi \lesssim 0.4\times10^{-10}$. In the trapping regime the bosons emerge
as quasi-thermal radiation from a region near the neutrino sphere and match
$L_\nu$ for $g_{a,\phi}\simeq 10^{-4}$. However, the $2\gamma$ decay is so fast
that all the energy is dumped into the surrounding progenitor-star matter,
whereas at most $10^{-2}E_{\rm SN}$ may show up in the explosion. To suppress
boson emission below this level we need yet larger couplings, $g_{a}\gtrsim
2\times10^{-3}$ and $g_{\phi}\gtrsim 4\times10^{-3}$. Muonic scalars can
explain the muon magnetic-moment anomaly for $g_{\phi}\simeq 0.4\times10^{-3}$,
a value hard to reconcile with SN physics despite the uncertainty of the
explosion-energy bound. For generic axion-like particles, this argument covers
the "cosmological triangle" in the $G_{a\gamma\gamma}$--$m_a$ parameter space.
