The moduli spaces of smooth projective curves with marked points is of fundamental importance in algebraic geometry and topology. However, only a tiny portion of its cohomology is understood. I will present joint works with Bibby, Chan and Yun, and with Hainaut, in which we gain access to the least algebraic part of the cohomology for curves of genus 2 and 3, using tropical geometry and configuration spaces of points on graphs. In particular we find the first examples of (large) families of cohomology classes in the top cohomological dimension, which seem to tell a geometric story that is yet to be understood.