  Debris discs are second generation dusty discs formed by collisions of
planetesimals. Many debris discs have been found and resolved around hot and
solar-type stars. However, only a handful have been discovered around M-stars,
and the reasons for their paucity remain unclear. Here we check whether the
sensitivity and wavelength coverage of present-day telescopes are simply
unfavourable for detection of these discs or if they are truly rare. We
approach this question by looking at the Herschel/DEBRIS survey that has
searched for debris discs including M-type stars. Assuming that these cool-star
discs are "similar" to those of the hotter stars in some sense (i.e., in terms
of dust location, temperature, fractional luminosity, or mass), we check
whether this survey should have found them. With our procedure we can reproduce
the $2.1^{+4.5}_{-1.7}$% detection rate of M-star debris discs of the DEBRIS
survey, which implies that these discs can indeed be similar to discs around
hotter stars and just avoid detection. We then apply this procedure to IRAM
NIKA-2 and ALMA bands 3, 6 and 7 to predict possible detection rates and give
recommendations for future observations. We do not favour observing with IRAM,
since it leads to detection rates lower than for the DEBRIS survey, with
0.6%-4.5% for a 15 min observation. ALMA observations, with detection rates
0.9%-7.2%, do not offer a significant improvement either, and so we conclude
that more sensitive far-infrared and single dish sub-millimetre telescopes are
necessary to discover the missing population of M-star debris discs.
