A point cloud data set, a dense set of discrete coordinate points scanned or sampled from the surface of a 3D physical object or design model, is emerging as a new representation format for geometric modeling. This paper presents a new method to detect tangential discontinuities in point cloud data. The method introduces an original criterion, named as incompatibility, to quantify the magnitude of shape change in the vicinity of a data point. The introduced criterion is unique since in smooth regions of the underlying surface where shape change around a data point is small, the calculated incompatibilities tend to cluster around small values. At points close to tangential discontinuities, the calculated incompatibilities become relatively large. By modeling the incompatibilities of points in smooth regions following a statistical distribution, the proposed method identifies tangential discontinuities as those points whose incompatibilities are considered outliers with respect to the distribution. As the categorization of outliers is in effect independent of the underlying surface shape and sampling conditions of the data points, a threshold can be automatically determined via a generic procedure and used to identify tangential discontinuities. The effectiveness of the proposed method is demonstrated through many case studies using both simulated and practical point cloud data sets.