The projection with centre an appropriate linear subspace contained in a cubic hypersurface has an associated discriminant hypersurface, which often has special properties such as many singularities. We study the geometry of linear subspaces contained in singular cubics and the Fano schemes parametrising them. This is done in the general setting and also for special cubics with many nodes, such as the Segre and Goryunov–Kalker cubics. From this we infer properties of the associated discriminants, especially in the two-dimensional setting, where we get surfaces with many nodes. For example, we construct a Togliatti quintic surface as a projection from a line on a 4-dimensional GK-cubic. We also show that the Togliatti quintic and the Barth sextic are unobstructed, and we apply representation theory to study the deformations of Segre and GK-cubic hypersurfaces in all dimensions, discussing some open questions.