  In this paper, we extend the method developed in [17, 18] 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.
