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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.

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The result is not true without additional assumptions. See A counterexample to the periodic orbit conjecture by Sullivan.

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.

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 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.

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    $\begingroup$ Million thanks! This is what I need (the rest is already found in Molino). $\endgroup$
    – Misha Verbitsky
    Commented Aug 12, 2021 at 7:30
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The question was already answered by Jorge Vitório Pereira, but let me add here what I have already found.

Recall that a foliation on a Riemannian manifold is called "Riemannian foliation" if the restriction of the Riemannian metric to the normal bundle is preserved by the holonomy of the foliation. The leaf space of a Riemannian foliation with compact leaves is an orbifold, a result which is probably due to Molino.

My question was related to the following question, which Sullivan calles "the periodic orbit conjecture". Suppose that $F$ is a rank foliation on a compact manifold with compact fibers. Is it true that there exists a circle action tangent to $F$?

The answer is positive in dimension 3 (D. B. A. Epstein, Periodic Flows on Three-Manifolds, Ann. M ath., 95 (1972), 66--82.) and false in dimension 5 (D. Sullivan, A counterexample to the periodic orbit conjecture Publications Mathématiques de l'IHÉS, Tome 46 (1976) , pp. 5-14.) When the foliation is Riemannian, the periodic flow conjecture is proven in the thesis of A. W. Wadsley, a student of Epstein (A. W. Wadsley, Geodesic foliations by circles, J. Differential Geometry {\bf 10} (1975), no. 4, 541--549.)

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    $\begingroup$ I assume, from your comments, that the conjecture is open in dimension four? If so, is this on some problem list someplace? (Because, it should be!) $\endgroup$
    – Sam Nead
    Commented Aug 12, 2021 at 12:06
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    $\begingroup$ @SamNead: I expanded my answer to include the counterexamples by Epstein and VVogt in dimension four. $\endgroup$
    – Jorge Vitório Pereira
    Commented Aug 12, 2021 at 12:42

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