Data in practical applications often contain structured discontinuities. This is, for example, the case if data stems from a physical process that exhibits shocks, such as in fracture mechanics, mass transport, or general phase-field problems such as generation of crystals or de-wetting. Also, in classification problems often encountered in machine learning, a decision boundary marks the interface between two or more regions. In this talk, we discuss the relevance of structured singularities in the functions that one attempts to learn. We will see that their shape directly affects the complexity of a learning problem. Moreover, we will observe that certain deep neural network-based algorithms can learn problems with complex but structured singularities at close to the statistically optimal rate. Moreover, we will identify cases where these learning rates are independent of the underlying dimension.