  We have used the Australia Telescope National Facility Mopra 22-m antenna to
search for 37.7-GHz (7(-2) - 8(-1}E) methanol masers towards a sample of thirty
six class II methanol masers. The target sources are the most luminous class II
methanol masers not previously searched for this transition, with isotropic
peak 12.2-GHz maser luminosity greater than 250 Jy/kpc^2 and isotropic peak
6.7-GHz maser luminosity greater than 800 Jy/kpc^2. Seven new 37.7-GHz methanol
masers were detected as a result of the search. The detection rate for 37.7-GHz
methanol masers towards a complete sample of all such class II methanol maser
sites south of declination -20 deg is at least 30 percent. The relatively high
detection rate for this rare methanol transition is in line with previous
predictions that the 37.7-GHz transition is associated with a late stage of the
class II methanol maser phase of high-mass star formation. We find that there
is a modest correlation between the ratio of the 6.7- and 37.7-GHz maser peak
intensity and the 6.7- and 12.2-GHz maser peak intensity (correlation
coefficient 0.63 in a log-log plot). We detected one new 38.3-GHz (6(2) -
5(3)A-) methanol maser towards G335.789+0.174. This is only the fourth source
for which maser emission has been detected in this transition and it is the
only one for which emission is not also observed in the 38.5-GHz 6(2) - 5(3)A+
transition.
