Is there a classification of symplectic surfaces, i.e. of surfaces equipped with an area form? Symplectic topology references like McDuff-Salamon seem to start their discussion of open questions with dimension four.


* A surface admits a symplectic form iff it is orientable.
* The Moser trick seems to show that on a *compact* orientable surface $M$, the unique invariant of a symplectic form is its total area? So the set of symplectic forms on $M$ (up to symplectomorphism) is parametrized by $\mathbb{R}^+$?
* And, for non-compact orientable surfaces, ...?



Classification of symplectic surfaces
This is a bit late answer to an old MO question, but Moser's theorem was generalized to open manifolds in 


R. Greene and K. Shiohama, Diffeomorphisms and volume-preserving embeddings of noncompact manifolds, Trans. Amer. Math. Soc. 255 (1979), p. 403-414.


<http://www.ams.org/journals/tran/1979-255-00/S0002-9947-1979-0542888-3/S0002-9947-1979-0542888-3.pdf>


They gave a necessary and sufficient condition for existence of volume-preserving diffeomorphisms between open manifolds in terms of volumes of ends and overall volumes. This takes care of surfaces as a very special case. 


