Combinatorics of T-singular surfaces

We describe concrete bounds that limit the number of possible types of T-singularies on a stable surface with a fixed number of singularities. While the ultimate bounds are complicated, they greatly simplify the search for T-singular surfaces. We focus on concrete examples of surfaces with multiple T-singularities (including something coming from a K3!) and discuss some of the combinatorial techniques used to obtain the bounds. This is joint work with Giancarlo Urzúa and Fernando Figueroa.