A vertex operator algebra $V$ is called simple if $V$ is a simple $V$-module. What are some examples of simple VOAs? Are there lots of examples or this is a very strong condition? Is there a classification? In particular I am interested if the following VOAs are simple or not and under what conditions:

1. The rank $d$ Heisenberg (free field) VOA
2. A lattice VOA of some non-degenerate lattice
3. Affine Lie algebra at level $k$ for a semisimple Lie algebra $\mathfrak{g}$
4. $\mathcal{W}_k(\mathfrak{g},f)$, $\mathcal{W}$ algebras associated to a semisimple Lie algebras at level $k$ and nilpotent element $f$. What if $\mathcal{W}_k(\mathfrak{g},f)$ is the principal $\mathcal{W}$-algebra?

A vertex operator algebra $V$ is called simple if $V$ is a simple $V$-module. What are some examples of simple VOAs? Are there lots of examples or this is a very strong condition? Is there a classification? In particular I am interested if the following VOAs are simple or not and under what conditions:

1. The rank $d$ Heisenberg (free field) VOA;
2. A lattice VOA of some non-degenerate lattice;
3. Affine Lie algebra at level $k$ for a semisimple Lie algebra $\mathfrak{g}$;
4. $\mathcal{W}_k(\mathfrak{g},f)$, $\mathcal{W}$ algebras associated to a semisimple Lie algebras at level $k$ and nilpotent element $f$. What if $\mathcal{W}_k(\mathfrak{g},f)$ is the principal $\mathcal{W}$-algebra?