Stationary varifolds generalize minimal surfaces and can exhibit singularities. The most general regularity theorem in this context is the celebrated Allard's Regularity Theorem, which asserts that the set of singular points has empty interior. However, it is believed that the set of singular points should have codimension (at least) one. Despite more than 50 years having passed since Allard's breakthrough, stronger results have remained elusive. In this talk, after a brief discussion about the regularity theory for stationary varifolds, I will discuss the principle of unique continuation and the topic of rectifiability, both of which are linked to understanding the structure of singularities. This discussion is based on joint works with Stefano Decio, Camillo De Lellis, and Federico Franceschini.