The quintic with 15 cusps is a so-called world record surface: To our knowledge, it is not known if there may be a quintic with more than 15 cusps although it is known that a quintic cannot have more than 20 cusps.
The Barth Sextic is probably the most famous example of the sometimes so-called world record surfaces. This post shows a smoothed variant of it.
A Togliatti quintic surface is a so-called world record surface. Among all quintic surfaces, it has the maximum possible number of singularities, namely 31. Our model is a smoothed version of such a surface.
Our modern version of Klein's historical cubic surface model with four singularities is the main figure in our photo from the series "math sculptures in context". It is the pure white version with its 9 straight lines.