  We find some equivalences of the derived category of coherent sheaves on a
Gorenstein genus one curve that preserve the (semi)-stability of pure
dimensional sheaves. Using them we establish new identifications between
certain Simpson moduli spaces of semistable sheaves on the curve. For rank
zero, the moduli spaces are symmetric powers of the curve whilst for a fixed
positive rank there are only a finite number of non-isomorphic spaces. We prove
similar results for the relative semistable moduli spaces on an arbitrary genus
one fibration with no conditions either on the base or on the total space. For
a cycle $E_N$ of projective lines, we show that the unique degree 0 stable
sheaves are the line bundles having degree 0 on every irreducible component and
the sheaves $\mathcal{O}(-1)$ supported on one irreducible component. We also
prove that the connected component of the moduli space that contains vector
bundles of rank $r$ is isomorphic to the $r$-th symmetric product of the
rational curve with one node.
