Let $M$ be a closed, orientable 3-manifold with a non-trivial differentiable $S^1$-action.
What does this imply for $M$? What are examples except for (products of) spheres?
You will find the complete answer in the papers
If the action has no fixed points, that class of manifolds has a name: Seifert-fibred 3-manifold with an orientation on the fiber. Seifert classified a slightly larger class of manifolds (Seifert-fibered ones -- no orientability constraint on the fibers). This is available in either the Jaco or Hatcher 3-manifolds lecture notes, also Orlik's book on Seifert-Fibered spaces.
Such examples include things like lens spaces, and relatively complicated 3-manifolds like the double of a torus knot complement.
When you allow fixed points, the orbit decomposition theorem gives you a stratification of the manifold into the fixed-point set (a link) together with its Seifert-fibred complement. So you could view these manifolds as certain Dehn Fillings on Seifert-fibred manifolds with boundary. As the references above point out, they're abundant.