  We give bounds for the cohomology of constructible sheaves on the moduli
stacks M_{g,n} over the complex field. This enables us recover Harer's bound
for the virtual cohomological dimension of the associated mapping class groups
as well the theorem of Diaz on complete subvarieties of M_g. We also obtain
such bounds for any open subset of the Deligne-Mumford compactification of
M_{g,n} that is a union of strata. Our proof yields a template for obtaining
similar bounds for the cohomological dimension for quasicoherent sheaves on
M_{g,n}.
