The curve graph is a well-studied and useful tool to study 3-manifolds and mapping class groups of surfaces. The fine curve graph is a recent variant on which the full homeomorphism group of a surface acts in an interesting way. In this talk we discuss some recent results which highlight behaviour not encountered in the 'classical' curve graph. In particular, we will discuss the first entries in a dictionary between properties from surface dynamics and geometric properties of the action (and, while doing so, construct homeomorphisms acting parabolically). This is joint work with Jonathan Bowden, Katie Mann, Emmanuel Militon and Richard Webb.[-]