Diagonal surface of Clebsch A cubic with 27 straight lines 
Cayley's Cubic A cubic with 4 A1 singularities

Cayley 1 A cubic with 3 A1+ singularities

Cayley 2 A cubic with a D4 singularity

Cayley 3 A cubic with an A2 singularity

Cayley 4 A cubic with a D6 singularity

E6 A cubic with an E6 singularity. The most singular cubic surface.

T[3,3,3] A cubic with a T[3,3,3] singularity

S20 A cubic with a line of A1 singularities

S21 A cubic with two crosscaps

S22 Yet another strange cubic

S23 One way to unfold a cusp

See also unfolding of three planes for some more cubics. 