Suppose that $M$ is a smooth manifold and $\mathfrak{X}(M)$ is the set of smooth vector fields on $M$. There are basically two different linear structures on $\mathfrak{X}(M)$:
1.) $\mathfrak{X}(M)$ is a (infinite dimensional) $\mathbb{R}$-vector space.
2.) $\mathfrak{X}(M)$ is an $C^\infty(M)$-module, where $C^\infty(M)$ means the algebra of smooth real valued functions on $M$.
(These structures are related by a so called Lie-Rinehart pair, but that's irrelevant for the question)
Now, the $C^\infty(M)$-dual of $\mathfrak{X}(M)$ is well known and precisely the $C^\infty(M)$-module of differential one-forms, that is
$$\Omega^1(M)=Hom_{C^\infty}(\mathfrak{X}(M),C^\infty(M))$$
The question is:
Is there moreover a common description of the $\mathbb{R}$-dual of $\mathfrak{X}(M)$? I mean, how can we think about the elements of
$$ Hom_\mathbb{R}(\mathfrak{X}(M),\mathbb{R}) $$
and are there places in mathematics where they appear?
I know this is pretty vague, but I'm just trying to 'get a hand' on this kind of dual.
Edit: From some of the comments/answers, it became clear to me, that there are better understood restrictions of $Hom_\mathbb{R}(\mathfrak{X}(M),\mathbb{R})$, so the question is generalized to
Is there a common description of (some meaningful vector subspace of) the $\mathbb{R}$-dual of $\mathfrak{X}(M)$?
First, by introducing a Riemannian metric, we get an isomorphism between the spaces of vector fields and (smooth) 1-forms on $M$. Thus, I will now work with 1-forms. In order to get a meaningful answer, I will restrict to the continuous dual of $\Omega^1(M)$, where continuity is defined as here. Then the dual space to $\Omega^1(M)$ is the space $D_1(M)$ of 1-currents on $M$. Examples of 1-currents are given by 1-chains, which are formal (real) linear combinations of smooth maps $\sigma: [0,1]\to M$ and duality is given by integration: $$ <\omega, \sum_{i=1}^k a_i \sigma_i > = \sum_{i=1}^k a_i \int_{\sigma_i} \omega. $$ Let me call such currents "polyhedral". Theory of currents is well-developed and you can find its most detailed treatment in the monumental book "Geometric Measure Theory" by H.Federer. In particular, under certain extra restrictions (rectifiability and integral multiplicity), Federer proves that every current can be approximated (in suitable topology) by polyhedral ones.
Federer's book is long and written in a very formal style where all the geometric intuition is left out. A much more gentle, geometric (and much less detailed) treatment can be found in F.Morgan's book "Geometric Measure Theory: A Beginner's Guide".
Lastly, for the dual question to yours (what is the "dual" to the space of polyhedral 1-currents) and my answer to it, see here.