Classification of Cubic Surfaces
In 1987, H. Knörrer and T. Miller, gave a classification of all real types of cubic surfaces using the language of moderne singularity theory. Our presentation of images and movies of these surfaces follows the classification described in their paper Mathematische Zeitschrift: Topologische Typen reeller kubischer Flächen.
Some of the lines on the surfaces are not appearing in a correct way, I will change this, but I hope that meanwhile the presentation will be sufficiant. Send me an e-mail if you are interested in a specific image or movie:

The equations for the surfaces where found using the two great papers Phil. Trans. Royal Soc.: A Memoire on Cubic Surfaces and Math. Ann., XIV: Zur Classification der Flächen dritter Ordnung written in the second half of the 19th century by A. Cayley and C. Rodenberg, respectively.

Rodenberg's article describes in a very explicit way not only how to write down good affine equations, but also explains in many cases how to deform a surfaces so that certain singularities occur and disappear.