"Classification" can mean more than one thing, but it's useful to be aware of the extensive development of *adjoint quotients* by Kostant, Steinberg, Springer, Slodowy, and others.   This makes sense for all semisimple (or reductive) groups and their Lie algebras over any algebraically closed field, perhaps avoiding a few small prime characteristics.   Older sources include Steinberg's 1965 IHES paper on regular elements and the Springer-Steinberg portion of the 
1970 Lecture Notes in Math. vol. 131.  (For an overview with references, based partly on Steinberg's Tata lectures, see Chapter 3 of my 1995 AMS book *Conjugacy Classes in Semisimple Algebraic Groups*.)   While the general focus has been on developing a picture of the collection of all classes or orbits as some kind of "quotient", quite a few special features of the classical types are also brought out in the Springer-Steinberg notes.   As suggested by Victor, Roger Carter's book *Finite Groups of Lie Type* also has a lot of related material but with special emphasis on nilpotent orbits.   The Jordan decomposition does reduce many classification questions to the nilpotent case, at least in principle, if you are willing to deal with various centralizers along the way.