  We discuss axion dark matter detection via two mechanisms: spontaneous decays
and resonant conversion in neutron star magnetospheres. For decays, we show
that the brightness temperature signal, rather than flux, is a less ambiguous
measure for selecting candidate objects. This is owing principally to the
finite beam width of telescopes which prevents one from being sensitive to the
total flux from the object. With this in mind, we argue that the large
surface-mass-density of the galactic centre or the Virgo cluster centre offers
the best chance of improving current constraints on the axion-photon coupling
via spontaneous decays. For the neutron star case, we first carry out a
detailed study of mixing in magnetised plasmas. We derive transport equations
for the axion-photon system via a controlled gradient expansion, allowing us to
address inhomogeneous mass-shell constraints for arbitrary momenta. We then
derive a non-perturbative Landau-Zener formula for the conversion probability
valid across the range of relativistic and non-relativistic axions and show
that the standard perturbative resonant conversion amplitude is a truncation of
this result in the non-adiabatic limit. Our treatment reveals that that
infalling dark matter axions typically convert non-adiabatically in
magnetospheres. We describe the limitations of one-dimensional mixing equations
and explain how three-dimensional effects activate new photon polarisations,
including longitudinal modes and illustrate these arguments with numerical
simulations in higher dimensions. We find that the bandwidth of the radio
signal is dominated by Doppler broadening from the relative motion of the
neutron star with respect to the observer. Therefore, we conclude that the
radio signal from the resonant decay is weaker than previously thought, which
means one relies on local density peaks to probe weaker axion-photon couplings.
