The detection of discontinuity curves of bivariate functions (and of their gradients) is an important problem that arises in many contexts, ranging from surface and scattered data reconstruction to edge detection in image and geometric processing. We present a fault detection method based on the so-called null rules, computed as a vector in the null space of certain collocation matrices. These rules are used as weights in a linear combination of function evaluations to indicate the local behavior of the function itself. By analyzing the asymptotic properties of the rules, we introduce two indicators (one for faults and one for gradient faults) by locally computing just one rule with degree of precision 2. This leads to a cheap and reliable scheme, which allows us to effectively detect and classify points close to discontinuities. We then show how this information can be suitably combined with adaptive approximation methods based on hierarchical spline spaces in the reconstruction process of surfaces with discontinuities. The considered adaptive methods exploit the ability of the hierarchical spaces to be locally refined, and the fault detection is a natural way to guide the refinement with low computational cost. Several numerical tests will be presented in order to show the behavior of the proposed methods.