  We consider log Calabi-Yau surfaces $(Y, D)$ with singular boundary. In each
deformation type, there is a distinguished surface $(Y_e,D_e)$ such that the
mixed Hodge structure on $H_2(Y \setminus D)$ is split. We prove that (1) the
action of the automorphism group of $(Y_e,D_e)$ on its nef effective cone
admits a rational polyhedral fundamental domain; and (2) the action of the
monodromy group on the nef effective cone of a very general surface in the
deformation type admits a rational polyhedral fundamental domain. These
statements can be viewed as versions of the Morrison cone conjecture for log
Calabi--Yau surfaces. In addition, if the number of components of $D$ is $\le
6$, we show that the nef cone of $Y_e$ is rational polyhedral and describe it
explicitly. This provides infinite series of new examples of Mori Dream Spaces.
