Structures which minimise area appear in numerous geometric contexts often related to degeneration phenomena. In turn, in many situations these structures also reflect the ambient geometry in some way (they are ‘calibrated’) and so they may provide a way to study the interplay between geometry and topology, as has historically been the case for variational methods in geometry. Almgren developed a theory which established that these area minimising structures are manifolds away from a codimension 2 ‘singular set’. The singular set itself, however, remained rather mysterious, including whether it necessarily has locally finite measure, unique tangent cones, or geometric structure (rectifiability). In this talk I will attempt to give an overview of these ideas, as well as of recent work (joint with Camillo De Lellis and Anna Skorobogatova) answering some of the questions above related to singularities of area minimisers.