  We study the action of the homeomorphism group of a surface $S$ on the fine
curve graph ${\mathcal C }^\dagger(S)$. While the definition of
$\mathcal{C}^\dagger(S)$ parallels the classical curve graph for mapping class
groups, we show that the dynamics of the action of ${\mathrm{Homeo}}(S)$ on
$\mathcal{C}^\dagger(S)$ is much richer: homeomorphisms induce parabolic
isometries in addition to elliptics and hyperbolics, and all positive reals are
realized as asymptotic translation lengths.
  When the surface $S$ is a torus, we relate the dynamics of the action of a
homeomorphism on $\mathcal{C}^\dagger(S)$ to the dynamics of its action on the
torus via the classical theory of rotation sets. We characterize homeomorphisms
acting hyperbolically, show asymptotic translation length provides a lower
bound for the area of the rotation set, and, while no characterisation purely
in terms of rotation sets is possible, we give sufficient conditions for
elements to be elliptic or parabolic.
