We study smooth foliations of compact manifolds where the set of noncompact leaves has zero Lebesgue measure (and is nonempty). We review a simple example due to Reeb and provide two new examples that show more complicated behavior that may occur.
"Essentially compact foliations that are not compact." Tohoku Math. J. (2) 56 (3) 317 - 326, 2004. https://doi.org/10.2748/tmj/1113246669