  We investigate the class of degenerations of smooth cubic surfaces which are
obtained from degenerating their Cox rings to toric algebras. More precisely,
we work in the spirit of Sturmfels and Xu who use the theory of Khovanskii
bases to determine toric degenerations of Del Pezzo surfaces of degree 4 and
who leave the question of classifying these degenerations in the degree 3 case
as an open problem. In order to carry out this classification we describe an
approach which is closely related to tropical geometry and present partial
results in this direction.
