Classification (and automorphisms) of torsion-free modules/sheaves I would like to know what can be said about the classification of torsion-free modules. 
For my purposes, we can assume that $R$ is the function ring of a smooth affine variety over a field. How does one proceed when trying to classify finitely generated torsion-free modules over $R$, or equivalently torsion-free sheaves on $\operatorname{Spec} R$? What general theorems are available, do there exist moduli spaces and what do they look like? 
I have seen that for $R=k[X_1,X_2]$, there is some relation between torsion-free sheaves on $\mathbb{A}^2$ and the Hilbert scheme of points on $\mathbb{A}^2$? Unfortunately, I have not found a precise statement. Where is the relation between isomorphism classes of torsion-free sheaves and points on the Hilbert scheme of points made explicit, and proved? Are there some classification results for torsion-free sheaves that generalize to affine spaces of arbitrary dimension, or even to smooth affine schemes in general? 
Finally, what can be said about automorphism groups of finitely generated torsion-free $R$-modules, automorphisms taken of course as $R$-modules? Are there any structural statements, either from the algebraic group or the discrete group point of view? Again, I would be interested both in the special case $k[X,Y]$ as well as in the case of smooth affine schemes.
[Edit: Thanks for the answer and comments I received so far. I would still like to know more about the automorphism groups of torsion-free modules. Are there any statements about this in the literature?]
 A: I expect that classification in general would be very difficult, but here are a few remarks about the simplest case. Let $M$ be a f.g. torsion free module over $R=k[x_1,x_2]$. Then to answer a question in your comment, although $M$ is not ideal in general, it does behave a bit like one.

Lemma. $M$ is a submodule of a free module in a canonical way.

The double dual $N=M^{**}$ is reflexive, which implies that it has depth $2$. Therefore, it follows from Auslander-Buchsbaum-Serre, that $N$ is projective and consequently free by (a very special case of) Quillen-Suslin. We has also have a canonical map $M\to N$, which is injective by torsion freeness, so the lemma follows.
This at least gives some kind of handle on $M$. Concretely, by choosing generators, you can see that $M$ is a image of a matrix.
As for moduli. If you take the sheaf associated to the quotient $N/M$ then this will be supported on a proper closed subscheme $Z$.
I would guess that you can parameterize the $N$'s where $Z$ is purely zero dimensional by a Quot scheme (generalizing the punctual Hilbert scheme). But I'm not sure that you can do anything like that in general, since the usual constructions require properness of the base. (I think you can get around it in previous case).
Added I never really thought about the automorphism group of an ideal before, but I think I can work out a simple example. Let $m=(x,y)$ be the maximal ideal of the origin. This can resolved by the Koszul complex $0 \to R\to R^2\to m$. From this it follows that $Aut(m)$ is the group of matrices in $GL_2(R)$ which stabilize the line spanned by $(-y,x)^T$
