  We perform a general study of the structure of locally compact modules over
compactly generated abelian groups. We obtain a devissage result for such
modules of the form "compact-by-sheer-by-discrete", and then study more
specifically the sheer part. The main typical example of a sheer module is a
polycontractable module, i.e., a finite direct product of modules, each of
which is contracted by some group element. We show that every sheer module has
a "large" polycontractable submodule, in a suitable sense.
  We apply this to the study of compactly generated metabelian groups. For
instance, we prove that they always have a maximal compact normal subgroup, and
we extend the Bieri-Strebel characterization of compactly presentable
metabelian groups from the discrete case to this more general setting.
