  In the near future, neutrino telescopes are expected to improve their
sensitivity to the flux of monochromatic neutrinos produced by dark matter (DM)
in our galaxy. This is illustrated by a new limit on the corresponding cross
section that we derive from public IceCube data. In this context, we study
which DM models could produce an observable flux of monochromatic neutrinos
from DM annihilations. To this end, we proceed in two steps. First, within a
set of simple and minimal assumptions concerning the properties of the DM
particle, we determine the models that could give rise to a significant
annihilation into monochromatic neutrinos at the freeze-out epoch. The list of
models turns out to be very limited as a result of various constraints, in
particular direct detection and neutrino masses at loop level. Given the fact
that, even if largely improved, the sensitivities will be far from reaching the
thermal annihilation cross section soon, a signal could only be observed if the
annihilation into neutrinos today is boosted with respect to the freeze-out
epoch. This is why, in a second step, we analyze the possibility of having such
a large enhancement from the Sommerfeld effect. For each scenario, we also
compute the cross sections into other annihilation products and confront our
results with experimental constraints. We find that, within our simple and
minimal assumptions, the expectation to observe monochromatic neutrinos is only
possible in very specific scenarios. Some will be confirmed or excluded in the
near future because they predict signals slightly below the current
experimental sensitivities. We also discuss how these prospects change by
relaxing our assumptions as well as by considering other types of sharp
spectral features. For the latter, we consider boxed-shaped and bremsstrahlung
spectra and provide the corresponding limits from IceCube data.
