Let $F$ be a smooth rank one foliation on a manifold $M$. Suppose that all leaves of $F$ are compact (that is, circles). Then its leaf space (edit: when additional assumptions are taken) is an orbifold. When the foliation is (transversally) Riemannian, this is proven in the book "Riemannian Foliations" by P. Molino. I suppose that this result is classical without the Riemannian assumption, but I could not find its proof in the literature. Any help with references is highly appreciated.



Smooth rank one foliations with closed leaves
The result is not true without additional assumptions. See [A counterexample to the periodic orbit conjecture](http://www.numdam.org/item/PMIHES_1976__46__5_0/) by Sullivan.




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**Added later.** The paper by Sullivan linked above exhibits a foliation by circles on a compact manifold of dimension $5$ with non-Hausdorff leaf-space. Later, Epstein and Vogt constructed a foliation by circles on a compact manifold of dimension $4$ with the same property, see [A Counterexample to the Periodic Orbit Conjecture in Codimension 3](https://doi.org/10.2307/1971187).


The problem makes sense for foliations of arbitrary dimensions. No need to be restricted to foliation by curves. There are also examples of holomorphic foliations having all its leaves compact but with non-Hausdorff leaf-space, on **non-compact** complex manifolds, see [On the stability of holomorphic foliations](https://link.springer.com/chapter/10.1007%2FBFb0097265) by Holmann.


As far as I know, there are no examples in the literature of holomorphic foliations with all its leaves compact and with non-Hausdorff leaf-space on a compact complex manifold.


