$S^1$-action in three dimensions 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?
 A: 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. 
A: You will find the complete answer in the papers
Frank Raymond, Classification of the actions of the circle on 3-manifolds. Trans. Amer. Math. Soc. 131 1968 51–78. 
and
Peter Orlik and Frank Raymond, Actions of SO(2) on 3-manifolds. 1968 Proc. Conf. on Transformation Groups (New Orleans, La., 1967) pp. 297–318 Springer, New York 
