In the answer of Will, one can exchange manifold $N$ to an orbifold. Then his answer works for effective action.

On the other hand assume you have a manifold $M$ and want to know if it admits an effective $\mathbb S^1$-action. In this case Will's answer is useless and as far as I know no good answer is known.

If there is an action on $M$ then

• There are some restrictions on $\pi_1(M)$; for example compact hyperbolic manifolds do not admit smooth $\mathbb S^1$-actions;
• If $M$ is simply connected and spin then $\hat A(M)$ has to vanish.

I do not know any other restriction.

In the answer of Will, one can exchange manifold $N$ to an orbifold with singularities modeled on $\mathbb R^{n-1}/\mathbb Z_k$ and $\mathbb R^{n}/\mathbb S^1$. Then his answer works for effective action. You can get a lot of examples this way, but I do not think you can call it a "classification".

On the other hand assume you have a manifold $M$ and want to know if it admits an effective $\mathbb S^1$-action. In this case Will's answer is useless and as far as I know no good answer is known.

If there is an action on $M$ then

• There are some restrictions on $\pi_1(M)$; for example compact hyperbolic manifolds do not admit smooth $\mathbb S^1$-actions;
• If $M$ is simply connected and spin then $\hat A(M)$ has to vanish.

I do not know any other restriction.