Which surfaces can be completely defined by a single parameterization?

It can be easily shown that any closed and bounded surface of $\mathbb{R}^3$ cannot be covered by a single surface patch, i.e. cannot be homeomorphic to an open set of $\mathbb{R}^2$. What can be said about the non-compact surfaces? Is it possible to characterize all the surfaces of $\mathbb{R}^3$ which are homeomorphic to an open set of $\mathbb{R}^2$?

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You're using many pieces of terminology informally and in ways that jar with the conventions used in geometry and topology, specifically the words "cover" and "classify" both probably do not mean what you intend in these fields. All surfaces have been classified, compact or non-compact. It's not too hard to specify which surfaces embed in $\mathbb R^3$ from that classification. For connected surfaces the result is a it embeds in $\mathbb R^3$ as long as it is not compact, boundaryless and containing a Moebius band. –  Ryan Budney Feb 4 at 6:59
May be 'characterize' is better, edited. And I guess the use of 'cover' is not much relevant in the view of my final question. –  pritam Feb 4 at 7:13
What do you want to know about the surfaced of $R^3$ which are homeomrphic to an open set of the plane? You pretty much said everything they have in common, I guess! –  Mariano Suárez-Alvarez Feb 4 at 7:34