In this article, we describe the structure of codimension one foliations with canonical singularities and numerically trivial canonical class on varieties with terminal singularities, extending a result of Loray, Pereira and Touzet to this context. Contents 1. Introduction 1 2. Notation, conventions, and used facts 4 3. Foliations 7 4. Singularities of foliations 10 5. Weakly regular foliations on singular spaces 17 6. Weakly regular foliations with algebraic leaves 25 7. Quasi-étale trivializable reflexive sheaves 28 8. A global Reeb stability theorem 30 9. Algebraic integrability, I 36 10. Algebraic integrability, II 40 11. Foliations defined by closed rational 1-forms 44 12. Proofs of Theorems 1.1 and 1.3 and proof of Corollary 1.4 51 References 55