Mechanical wave transmission through a material is influenced by the mechanical discontinuity in the material. The propagation of embedded discontinuities can be monitored by analyzing the wave-transmission measurements recorded by a multipoint sensor system placed on the surface of the material. In our study, robust monitoring of the propagation of a mechanical discontinuity is achieved by using supervised learning followed by data-driven causal discovery to process the multipoint waveform measurements resulting from a single impulse source. The new data-driven causal-discovery workflow jointly processes the nine 25-µs waveforms measured by the multipoint sensor system comprising 9 sensors. The proposed workflow can monitor the propagation of mechanical discontinuity through three stages, namely initial, intermediate, and final stages. The workflow considers the wave attenuation, dispersion and multiple wave-propagation modes. Among various feature reduction techniques ranging from decomposition methods to manifold approximation methods, the features derived based on statistical parameterizations of the measured waveforms lead to reliable monitoring that is robust to changes in precision, resolution, and signal-to-noise ratio of the multipoint sensor measurements. Causal signatures have been successfully identified in the multipoint waveform measurements. The numbers of zero-crossing, negative-turning, and positive in the waveforms are the strongest causal signatures of crack propagation. Higher order moments of the waveforms, such as variance, skewness and kurtosis, are also strong causal signatures of crack propagation. The newly discovered causal signatures confirm that the statistical correlations and conventional feature rankings are not always statistically significant indicators of causality.