  We investigate the coarse homology of leaves in foliations of compact
manifolds. This is motivated by the observation that the non-leaves constructed
by Schweitzer and by Zeghib all have non-finitely generated coarse homology.
This led us to ask whether the coarse homology of leaves in a compact manifold
always has to be finitely generated. We show that this is not the case by
proving that there exist many leaves with non-finitely generated coarse
homology. Moreover, we improve Schweitzer's non-leaf construction and produce
non-leaves with trivial coarse homology.
