  In this paper, we study Higgs and co-Higgs bundles on non-K\"ahler elliptic
surfaces. We show, in particular, that non-trivial stable Higgs bundles only
exist when the base of the elliptic fibration has genus at least two and use
this existence result to give explicit topological conditions ensuring the
smoothness of moduli spaces of stable rank-2 sheaves on such surfaces. We also
show that non-trivial stable co-Higgs bundles only exist when the base of the
elliptic fibration has genus 0, in which case the non-K\"ahler elliptic surface
is a Hopf surface. We then given a complete description of non-trivial co-Higgs
bundles in the rank 2 case; these non-trivial rank-2 co-Higgs bundles are
examples of non-trivial holomorphic Poisson structures on
$\mathbb{P}^1$-bundles over Hopf surfaces.
