Here is a a surface of degree 6 (a *sextic*)
in projective three-space
that has 65 nodes.
It was discovered by
W. Barth
in 1996.
The amazing fact here is that
a sextic can never have
*more*
than 65 nodes
--
so Barth's
sextic is really an extreme case.
It is in fact
the first known example of this kind.
Look into the paper

- Barth, W.: Two projective surfaces with many nodes, admitting the symmetries of the icosahedron. J. Algebraic Geometry 5, 173-186 (1996)

The nice computer picture shown above was made by Stephan Endrass using his program SURF. (SURF can draw pictures of a surface in three-space when you provide an equation of the surface.)

If you are interested in a classical surface, don't forget to have a look at Clebsch's diagonal cubic.