Abstract: In this work, we extend the method developed in [SALARINOGHABI, M.; TARI, F. Flat and round singularity theory of plane curves. Q. J. Math., v. 68, n. 4, p. 1289–1312, 2017] to curves in the Minkowski plane. The method proposes a way to study deformations of plane curves taking into consideration their geometry as well as their singularities. We deal in detail with all local phenomena that occur generically in 2-parameters families of curves. In each case, we obtain the geometry of the deformed curve, that is, information about inflections, vertices and lightlike points. We also obtain the behavior of the evolute/caustic of a curve at especial points and the bifurcations that can occur when the curve is deformed. Moreover, in order to obtain the generic deformations at a lightlike inflection point of order 2, we also classify submersions from  to  by diffeomorphisms in the source that preserve the swallowtail and, using such classification, we study the flat geometry of the swallowtail, which comes from its contact with planes, which in turn is measured by the singularities of the height function on the swallowtail.