  We study torsion-free, rank 2 Higgs sheaves on genus one fibered surfaces,
(semi)stable with respect to suitable polarizations in the sense of Friedman
and O'Grady. We prove that slope-semistability of a Higgs sheaf on the surface
implies semistability on the generic fiber. In the case of Higgs sheaves of odd
fiber degree on elliptic surfaces in characteristic $\neq 2$, we prove that any
moduli space of Higgs sheaves with fixed numerical invariants splits
canonically as the product of the moduli space of ordinary sheaves (with the
same invariants), and the space of global regular $1$-forms on the surface. For
elliptic surfaces with section in characteristic zero, and in the case
arbitrary fiber degree, we prove that if a Higgs sheaf has reduced Friedman
spectral curve, or is regular on a general fiber with non-reduced spectral
cover, then its Higgs field takes values in the saturation of the pull-back of
the canonical bundle of the base curve in the cotangent bundle of the surface.
