Embedded leaves need not be closed and leaves need not be embedded

On the annulus $A=\{(r,\theta ):1\mathrm{/}2<r<3\mathrm{/}2\}$, the vector field [given] $X={\mathrm{\partial }}_{\theta }+(r-1{)}^2{\mathrm{\partial }}_r$ is nowhere zero, so its integral curves form a regular one-dimensional foliation. The circle $r=1$ is one leaf, and every nearby noncircular leaf is a non-self-intersecting spiral whose closure contains that circle. Such a spiral leaf is embedded but not closed in $A$.

On the torus ${\mathbb{T}}^2={\mathbb{R}}^2\mathrm{/}{\mathbb{Z}}^2$, the constant vector [given] field generated by $(1,\alpha )$ with irrational $\alpha$ gives a regular foliation by injectively immersed images of $\mathbb{R}$. Each leaf is dense in ${\mathbb{T}}^2$, hence cannot be embedded.

Therefore global Frobenius theory correctly concludes only that leaves are [given] injectively immersed; neither closedness nor embeddedness is automatic.