All (not only finite dimensional!!) irreducible reps of $W$-algebra of $sl_2$ at its minimal nilpotent (read $U(sl_2)$) were classified by Block. A similar classification for the minimal $W$-algebra of $sl_3$ is feasible but still awaits its hero, to the best of my knowledge.

All (not only finite dimensional!!) irreducible reps of $W$-algebra of $sl_2$ at its subregular nilpotent (read $U(sl_2)$) were classified by Block. A similar classification for the minimal $W$-algebra of $sl_3$ is feasible but still awaits its hero, to the best of my knowledge.