  The new generation of powerful instruments is reaching sensitivities and
temporal resolutions that will allow multi-messenger astronomy of explosive
transient phenomena, with high-energy neutrinos as a central figure. We derive
general criteria for the detectability of neutrinos from powerful transient
sources, for given instrument sensitivities. In practice, we provide the
minimal photon flux necessary for neutrino detection based on two main
observables, the bolometric luminosity and the time variability of the
emission. This limit can be compared to the observations in specified
wavelengths in order to target the most promising sources for follow-ups. Our
criteria can also help discriminating false associations of neutrino events
with a flaring source. We find that relativistic transient sources such as
High- and Low-Luminosity GRBs, Blazar flares, Tidal Disruption Events and
magnetar flares could be observed with IceCube, as they have a good chance to
occur within a detectable distance. Among non-relativistic transient sources,
only luminous supernovae appear as promising candidates. We caution that our
criterion should not be directly applied to low-luminosity GRBs and Type Ibc
supernovae, as these objects could have hosted a choked GRB, leading to
neutrino emission without a relevant counterpart radiation. We treat a set of
concrete examples and show that several transients, some of which are being
monitored by IceCube, are far from meeting the criterion for detectability
(e.g., Crab flares or Swift J1644+57).
