biplanar double point | conic double point | elliptic cone point | isolated singular point | uniplanar double point

An isolated singularity is a singularity for which there exists a (small) real number ϵ such that there are no other singularities within a neighborhood of radius ϵ centered about the singularity. Isolated singularities are also known as conic double points. The types of isolated singularities possible for cubic surfaces have been classified (Schläfli 1863, Cayley 1869, Bruce and Wall 1979) and are summarized in the following table from Fischer.

cubic surface | rational double point | singular point