  We report the results of 870-$\mu$m continuum observations, using the Large
APEX Bolometer Camera (LABOCA), towards 77 class-II, 6.7-GHz methanol masers
identified by the Methanol Multibeam (MMB) survey to map the thermal emission
from cool dust towards these objects. These data complement a study of 630
methanol masers associated with compact dense clumps identified from the
ATLASGAL survey. Compact dust emission is detected towards 70 sources, which
implies a dust-association rate of 99% for the full MMB catalogue. Evaluation
of the derived dust and maser properties leads us to conclude that the combined
sample represents a single population tracing the same phenomenon. We find
median clump masses of a few 10$^3$ M$\odot$ and that all but a handful of
sources satisfy the mass-size criterion required for massive star formation.
This study provides the strongest evidence of the almost ubiquitous association
of methanol masers with massive, star-forming clumps. The fraction of
methanol-maser associated clumps is a factor of ~2 lower in the outer Galaxy
than the inner Galaxy, possibly a result of the lower metallicity environment
of the former. We find no difference in the clump-mass and maser-luminosity
distributions of the inner and outer Galaxy. The maser-pumping and
clump-formation mechanisms are therefore likely to be relatively invariant to
Galactic location. Finally, we use the ratio of maser luminosity and clump mass
to investigate the hypothesis that the maser luminosity is a good indicator of
the evolutionary stage of the embedded source, however, we find no evidence to
support this.
