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.