  In this paper, we suggest a new approach to study minimal surfaces of general
type with $p_g=0$ via their Cox rings, especially using the notion of
combinatorially minimal Mori dream space introduced by Hausen. First, we study
general properties of combinatorially minimal Mori dream surfaces. Then we
discuss how to apply these ideas to the study of minimal surfaces of general
type with $p_g=0$ which are very important but still mysterious objects. In our
previous paper, we provided several examples of Mori dream surfaces of general
type with $p_g=0$ and computed their effective cones explicitly. In this paper,
we study their fibrations, explicit combinatorially minimal models and discuss
singularities of the combinatorially minimal models. We also show that many
minimal surfaces of general type with $p_g=0$ arise from the minimal
resolutions of combinatorially minimal Mori dream surfaces.
