In this paper, we introduce a natural notion of constant curvature Lorentzian surfaces with conical singularities, and provide a large class of examples of such structures. We moreover initiate the study of their global rigidity, by proving that de-Sitter tori with a single singularity are essentially determined by the topological equivalence class of their lightlike bi-foliation. While this result is reminiscent of Troyanov's work on Riemannian surfaces with conical singularities, the rigidity will come from topological dynamics in the Lorentzian case.