The thin obstacle problem is a classical free boundary problem arising from the study of an elastic membrane resting on a lower-dimensional obstacle. Concerning the behavior of the solution near a contact point between the membrane and the obstacle, many important questions remain open. In this talk, we discuss a unified method that leads to a rate of convergence to `tangent cones’ at contact points with integer frequencies in general dimensions as well as 7/2-frequency points in 3d. This talk is based on recent joint works with Ovidiu Savin (Columbia).