I'm hoping that the following are true.  In fact, they are probably easy, but I'm not seeing the answers immediately.

Let $M$ be a smooth $m$-dimensional manifold with chosen positive smooth density $\mu$, i.e. a chosen (adjectives) volume form.  (A _density_ on $M$ is a section of a certain trivial line bundle.  In local coordinates, the line bundle is given by the transition maps $\tilde\mu = \left| \det \frac{\partial \tilde x}{\partial x} \right| \mu$.  When $M$ is oriented, this bundle can be identified with the top exterior power of the cotangent bundle.)  __Hope 1:  Near each point in $M$ there exist local coordinates $x: U \to \mathbb R^m$ so that $\mu$ pushes forward to the canonical volume form $dx$ on $\mathbb R^m$.__

Hope 1 is certainly true for volume forms that arise as top powers of symplectic forms, for example, by always working in Darboux coordinates.  If Hope 1 is true, then $M$ has an atlas in which all transition maps are volume-preserving.  My second Hope tries to describe these coordinate-changes more carefully.

Let $U$ be a domain in $\mathbb R^m$.  Recall that a change-of-coordinates $\tilde x(x): U \to \mathbb R^m$ is _oriented-volume-preserving_ iff $\frac{\partial \tilde x}{\partial x}$ is a section of a trivial ${\rm SL}(n)$ bundle on $U$.  An _infinitesimal change-of-coordinates_ is a vector field $v$ on $U$, thought of as the map $x \mapsto x + \epsilon v(x)$.  An infinitesimal change-of-coordinates is necessarily orientation-preserving; it is volume preserving iff $\frac{\partial v}{\partial x}(x)$ is a section of a trivial $\mathfrak{sl}(n)$ bundle on $U$.  __Hope 2:  The space of oriented-volume-preserving changes-of-coordinates is generated by the infinitesimal volume-preserving changes-of-coordinates, analogous to the way a finite-dimensional connected Lie group is generated by its Lie algebra.__

Hope 2 is not particularly well-written, so Hope 2.1 is that someone will clarify the statement.  Presumably the most precise statement uses infinite-dimensional Lie groupoids.  The point is to show that a certain _a priori_ coordinate-dependent construction in fact depends only on the volume form by showing that the infinitesimal changes of coordinates preserve the construction.

<b><i>Edit:</i></b> I have preciseified Hope 2 as [this question](https://mathoverflow.net/questions/7853/is-the-space-of-volume-preserving-maps-path-connected).