  We study the impact of neutrino magnetic moments on astrophysical neutrinos,
in particular supernova neutrinos and ultra-high energy neutrinos from
extragalactic sources. We show that magnetic moment-induced conversion of
left-handed neutrinos into unobservable right-handed singlet states can
substantially change the flux and flavour composition of these neutrinos at
Earth. Notably, neutrinos from a supernova's neutronisation burst, whose flux
can be predicted with O(10%) accuracy, offer a discovery reach to neutrino
magnetic moments $\sim \text{few} \times 10^{-13} \mu_B$, up to one order of
magnitude below current limits. For high-energy neutrinos from distant sources,
for which no robust flux prediction exists, we show how the flavour composition
at Earth can be used as a handle to establish the presence of non-negligible
magnetic moments, potentially down to $\text{few} \times 10^{-17} \mu_B$ if the
measurement can be performed on neutrinos from a single source. In both cases,
the sensitivity strongly depends on the galactic resp. intergalactic magnetic
field profiles along the line of sight. Therefore, while a discovery is
possible down to very small values of the magnetic moment, the absence of a
discovery does not imply an equally strong limit. We also comment on the
dependence of our results on the right-handed neutrino mass, paying special
attention to the transition from coherent deflection by a classical magnetic
field to incoherent scattering on individual scattering targets. Finally, we
show that a measurement of Standard Model Dirac neutrino magnetic moments, of
order $10^{-19} \mu_B$, could be possible under rather optimistic, but not
completely outrageous, assumptions using flavour ratios of high-energy
astrophysical neutrinos.
