The answer to the general question is "no":
If $\mathfrak{g}$ is solvable, by Lie's theorem its commutant $\mathfrak{g}^{\prime}=[\mathfrak{g},\mathfrak{g}]$ is represented by strictly upper triangular matrices in a suitable basis in any finite-dimensional module. Hence all "trace generated" polynomials are zero on $\mathfrak{g}^{\prime}$; in other words, they factor through the abelianization $\mathfrak{g}/\mathfrak{g}^{\prime}$ and are generated by linear invariant polynomials. Unless the adjoint action of $G$ with Lie algebra $\mathfrak{g}$ on $\mathfrak{g}^{\prime}$ has a Zariski dense orbit, there are invariant polynomials that cannot be obtained in this way.
The answer to the claim is "yes", this is Kostant's theorem from his celebrated paper:
If $G$ is a complex semisimple group then its nullcone $\mathcal{N}\subset\mathfrak{g}$ is the Zariski closure of a single adjoint orbit consisting of regular nilpotent elements.
Kostant actually proved that the nullcone is the scheme-theoretic complete intersection defined by $rk\;G$ homogeneous positive degree algebra generators of $\mathbb{C}[\mathfrak{g}]^G$ — this is the connection with the Chevalley theorem mentioned by others. But for the present purpose, it is enough to show that regular nilpotents are Zariski open and dense in $\mathcal{N}\cap\mathfrak{n},$ and a good way of doing it was indicated by David Speyer.
Here is answer (YES) from Alexey Bolsinov who is one the main experts in these questions.
"The answer is YES
There is a very general construction allowing to construct an integrable system on more or less any coadjoint orbit for an arbitrary Lie algebra (non necessarily semisimple). This is a recent paper by Vinberg and Yakimova available in arxiv
http://arxiv.org/abs/math/0511498
Complete families of commuting functions for coisotropic Hamiltonian actions
In the particular case you are talking about (SEMI-SIMPLE g) the positive answer follows from 2 results:
1) the so-called shifts of polynomial invariants give a completely integrable system on a singular adjoint orbit O(b) in a semi simple Lie algebra G if and only if
the index of the centralizer of b coincides with the index of G
(my paper in Izvestija AN SSSR, 1991 and Acta Appl. Math. 1991), both available on my home page
Bolsinov A.V. Commutative families of functions related to consistent Poisson brackets// Acta Appl. Math., 24(1991), pp. 253-274.
I also conjectured that
this condition ind Cent (b) = ind G, in fact, holds true for all singular elements b\in G and checked it for G=sl(n) (in particular for all nilpotent)
2) This conjecture (widely known as Elashvili conjecture) has been proved for an arbitrary semi simple Lie algebra and for all elements (in fact the proof is easily reduced to nilpotent elements)
First, Elashvili did it by, in some sense, straightforward computation which in the most difficult case of e_8 involved some computer program (unpublished)
Recently a conceptual proof has been done by Jean-Yves Charbonnel (IMJ), Anne Moreau (available in arxiv)
http://arxiv.org/abs/1005.0831
The index of centralizers of elements of reductive Lie algebras
To the best of my knowledge, this is the only known universal way to construct an integrable system on an arbitrary orbit.
Remark: I am talking about classical integrable systems, not quantum. These systems can be quantized too, but this is another story.
"
Best Answer
"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:
(MSN and article.) Their references include the work by Burgoyne–Cushman, Milnor, Springer-Steinberg mentioned by Bruce and me.