"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 (MSN and article) and the Springer–Steinberg portion of the
1970 Lecture Notes in Math. vol. 131 (MSN and chapter). (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 (MSN, book, and chapter).) 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 (MSN) 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.
[ADDED] The paper in J. Math. Physics linked below gives a nice concrete answer to the original question, building on some of the older theory but using mainly tools from linear algebra and basic group theory. This is the traditional approach of most physicists, though papers in this mixed journal are sometimes unreliable and contain mathematics of the sort probably not usable in physics but also not publishable in math journals. Djokovic and his collaborators are more reliable than most, fortunately, and he has written many papers using parts of Lie theory as well. One downside is the narrower perspective than found in the notes of Springer–Steinberg, for instance. But it all depends on whether you want to work over other fields or want to organize the classes/orbits more conceptually.
Here is a MathSciNet reference:
MR708648 (85g:15018) 15A21 (17B99 17C99)
Djokovic´,D. Zˇ . [¯Dokovic´, Dragomir Zˇ .] (3-WTRL); Patera, J. [Patera, Jirˇ´ı] (3-MTRL-R);
Winternitz, P. (3-MTRL-R); Zassenhaus, H. (1-OHS)
Normal forms of elements of classical real and complex Lie and Jordan algebras.
J. Math. Phys. 24 (1983), no. 6, 1363–1374 (review by R.C. King).
(MSN and article.) Their references include the work by Burgoyne–Cushman, Milnor, Springer-Steinberg mentioned by Bruce and me.