In this talk, I will present two results relating the qualitative dynamics of non-degenerate Hamiltonian isotopies on surfaces to the structure of their Floer complexes.   The first will be a topological characterization of those Floer chains which represent the fundamental class in CF∗(H,J) and which moreover lie in the image of some chain-level PSS map. This leads to a novel symplectically bi-invariant norm on the group of Hamiltonian diffeomorphisms, which is both C0-continuous and computable in terms of the underlying dynamics. The second result explains how certain portions of the Hamiltonian Floer chain complex may be interpreted geometrically in terms of positively transverse singular foliations of the mapping torus, with singular leaves given by certain maximal collections of unlinked orbits of the suspended flow. This construction may be seen to provide a Floer-theoretic construction of the `torsion-low’ foliations which appear in Le Calvez’s theory of transverse foliations for surface homeomorphisms, thereby establishing a bridge between the two theories.