The answer may depend on what you mean by "classified", but the best results so far seem to be those of Brundan-Kleshchev and Brown-Goodwin. See for example the preprint here and its references. (However, the Brown-Goodwin results might not apply directly to your situation, due to their restrictive assumption on the nilpotent orbit.)

ADDED: I'm far from being an expert on the subject, but maybe I can focus the discussion a little more. The notion of finite $W$-algebra is tricky to define (there being at least three equivalent but different-looking definitions in the literature). Essentially it's an associative algebra attached to a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and a nilpotent element $e \in \mathfrak{g}$. Usually the case $e=0$ is ignored, but the convention then is that the resulting finite $W$-algebra is just the universal enveloping algebra $U(\mathfrak{g})$. Kostant originally studied the opposite extreme when $e$ is regular ("principal") and the resulting algebra is isomorphic to the center of $U(\mathfrak{g})$. Premet extended the definition, using an $\mathfrak{sl}_2$-triple for (nonzero) $e$.

In any case, the finite dimensional representations are the main object of study; even their existence is problematic at first. Only in rank 1 are the results given by classical methods, while in general there are close connections with the primitive ideals of the universal enveloping algebra. Besides the work I mentioned above I should have pointed out the papers by Premet and by Ivan Losev: see for example Losev's recent ICM talk here and his earlier paper here.

The answer may depend on what you mean by "classified", but the best results so far seem to be those of Brundan-Kleshchev and Brown-Goodwin. See for example the preprint here and its references. (However, the Brown-Goodwin results might not apply directly to your situation, due to their restrictive assumption on the nilpotent orbit.)

ADDED: I'm far from being an expert on the subject, but maybe I can focus the discussion a little more. The notion of finite $W$-algebra is tricky to define (there being at least three equivalent but different-looking definitions in the literature). Essentially it's an associative algebra attached to a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and a nilpotent element $e \in \mathfrak{g}$. Usually the case $e=0$ is ignored, but the convention then is that the resulting finite $W$-algebra is just the universal enveloping algebra $U(\mathfrak{g})$. Kostant originally studied the opposite extreme when $e$ is regular ("principal") and the resulting algebra is isomorphic to the center of $U(\mathfrak{g})$. Premet extended the definition, using an $\mathfrak{sl}_2$-triple for (nonzero) $e$.

In any case, the finite dimensional representations are the main object of study; even their existence is problematic at first. Only in rank 1 are the results given by classical methods, while in general there are close connections with the primitive ideals of the universal enveloping algebra. Besides the work I mentioned above I should have pointed out the papers by Premet and by Ivan Losev: see for example Losev's recent ICM talk here and his earlier paper here.

The answer may depend on what you mean by "classified", but the best results so far seem to be those of Brundan-Kleshchev and Brown-Goodwin. See for example the preprint here and its references. (However, the Brown-Goodwin results might not apply directly to your situation, due to their restrictive assumption on the nilpotent orbit.)

The answer may depend on what you mean by "classified", but the best results so far seem to be those of Brundan-Kleshchev and Brown-Goodwin. See for example the preprint here and its references. (However, the Brown-Goodwin results might not apply directly to your situation, due to their restrictive assumption on the nilpotent orbit.)

ADDED: I'm far from being an expert on the subject, but maybe I can focus the discussion a little more. The notion of finite $W$-algebra is tricky to define (there being at least three equivalent but different-looking definitions in the literature). Essentially it's an associative algebra attached to a semisimple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$ and a nilpotent element $e \in \mathfrak{g}$. Usually the case $e=0$ is ignored, but the convention then is that the resulting finite $W$-algebra is just the universal enveloping algebra $U(\mathfrak{g})$. Kostant originally studied the opposite extreme when $e$ is regular ("principal") and the resulting algebra is isomorphic to the center of $U(\mathfrak{g})$. Premet extended the definition, using an $\mathfrak{sl}_2$-triple for (nonzero) $e$.

In any case, the finite dimensional representations are the main object of study; even their existence is problematic at first. Only in rank 1 are the results given by classical methods, while in general there are close connections with the primitive ideals of the universal enveloping algebra. Besides the work I mentioned above I should have pointed out the papers by Premet and by Ivan Losev: see for example Losev's recent ICM talk here and his earlier paper here.