Given a smooth, non-vanishing vector field on a compact manifold, when does the 1-dimensional foliation given by its integral curves admit a transverse invariant measure?
I've seen examples of higher-dimensional foliations not admitting transverse invariant measures, but I'd imagine the same question is much easier to address one way of the other in the one-dimensional case.
I think that the best reference for this question is still the (relatively) old paper by Plante Foliations with measure preserving holonomy Ann. of Math. (2) 102 (1975), no. 2, 327–361, although it is a bit of an overkill for one-dimensional foliations. For instance, by Theorem 4.1 holonomy invariant measures exist for any foliation with a subexponential leaf. There are many other ways, of course.
As R W says, the answer is in Plante's paper, but the special case of $1$-dimensional foliations is actually simple, and well-known: let $M$ be the manifold, $*\in M$ be a basepoint, $(\phi^t)$ be the flow (assumed tangential to the boundary of $M$, if any). For every integer $n\ge 0$, let $\mu_n$ be the image in $M$ of the probability measure $dt/(2n)$ on the interval $[-n,+n]$ under the map $$F:R\to M:t\mapsto\phi^t(*)$$ Since the space of Borelian probability measures on $M$ is compact for the weak topology, the sequence $(\mu_n)$ has a subsequence weakly converging to a Borelian probability measure $\mu$ on $M$. Claim: $\mu$ is invariant by every $\phi^t$. Indeed, for a fixed $t$, the sequence of measures $$\phi^t_*\mu_n-\mu_n$$ goes to $0$ in the norm topology, since for any continuous real function $f$ on $M$, one has: $$\vert(\phi^t_*\mu_n-\mu_n)f\vert=\frac{1}{2n}\vert\int_{-n+t}^{n+t}f(\phi^s(*))ds-\int_{-n}^{+n}f(\phi^s(*))ds\vert\le\frac{t}{n}\Vert f\Vert_\infty$$ Finally, the $R$-invariant measure $\mu$ amounts to a transverse invariant measure $\nu$ such that locally (in every local flow box $B\cong D^{n-1}\times I$) one has $\mu=\nu\otimes dt$. The measure $\nu(D)$ of every small transverse disk $D$ is the limit of some subsequence of the sequence $$\frac{1}{2n}{\sharp(F^{-1}(D)\cap[-n,+n])}$$ In this construction, we have used the choice axiom (hidden in the compacity of the measures space); in particular, the construction of the subsequence is non-constructive.
