Reeb's global stability theorem requires the foliation to be of codimension 1. As a counterexample, in "Geometric theory of foliations", Camacho and Lins Neto present the following.
Consider the manifold $S^{n-2}\times S^1\times S^1$ with coordinates $(x_1,...,x_{n-1},\varphi,\theta)$ such that $\sum_{i=1}^{n-1}x_i^2 = 1$ , and two differential one-forms
$$\eta_1 = d\theta \quad\text{ and }\quad\eta_2= ((1-\sin(\theta))^2 + x_1^2) d\varphi + \sin(\theta) dx_1 .$$
Frobenius' integrability condition is satisfied as can be easily checked, so there's an integral manifold of codimension two for these differential forms.
There are some compact leaves homeomorphic to $S^{n-2}$ given by fixing $\theta = 0$ and $\varphi = \mathit{constant}$.
On the other hand, there are leaves of a second kind given by $\theta = \pi/2$ and $\varphi = \mathit{constant} + \frac{1}{x_1}$ and these are said to be non-compact.
Question 1: Is there any simple argument for non-compactness of these leaves?
Intuition: Some sequence in one of these leaves for which the first coordinate converges to zero, cannot converge in the same leaf "because it is not given by the same equation ($\varphi = \frac{1}{x_1}$)". I'm not sure if this is correct, and even if it was, I couldn't find some neat way to explain it.
Question 2: We have that $d\eta_2 = 2x_1 dx_1\wedge d\varphi $ on the leaves of the second kind but is zero when $x_1$ is. Does this give some topological difference for leaves?