We propose a statistical criterion of digital curves regularity. It is not defined from an a priori model but a contrario to some statistics on random walks. This allows to automatically define detection thresholds in terms of a false detection rate. We apply this algorithm to the level lines of gray level images and experimentally check the statement of the Gestalt Theory following which regularity makes curve conspicuous without any contrast information, and that edges are really often good continuations. We also define good continuations breakings, which are good candidates for T-junctions, corners and Julesz's terminators. We also show that detection is not improved by shape scale space.