In 1973 Schaeffer established a result that applies to scalar conservation laws with convex fluxes and can be loosely speaking formulated as follows: for a generic smooth initial datum, the admissible solution is smooth outside a locally finite number of curves in the (t,x) plane. Here the term ”generic” should be interpreted in a suitable technical sense, related to the Baire Category Theorem. My talk will aim at discussing a recent explicit counter-example that shows that Schaeffer’s Theorem does not extend to systems of conservation laws. The talk will be based on joint works with Laura Caravenna.