This should literally be a textbook example of how phrasing and context affect polling

Talking Points Memo discusses how two polls given in the same week can get starkly different results:

Last week, a Gallup poll showed that 60% of Americans would rather see a budget compromise than see members of Congress who represent their interests hold out for their ideal budget, if it means the government would shut down. That phrases the current debate in Washington fairly concisely.

Compare that to Rasmussen, which framed the question much differently:

5* Would you rather have Congress avoid a government shutdown by authorizing spending at the same levels as last year or would you rather have a partial government shutdown until Democrats and Republicans can agree on what spending to cut?

That frames the budget showdown as an either or: either the government continues spending at current levels, or it shuts down until cuts are made. In response to that, 58% of likely voters said they preferred a government shutdown.

Rasmussen is, of course, something of a special case. To understand just how special, take a look at Nate Silver's excellent analysis,"When ‘House Effects’ Become ‘Bias.’"