Microsoft recently announced that solving this problem is now as easy as typing the song name and "lyrics" into Bing. The search engine will then show the complete lyrics for the song in its search results page.
To start, Microsoft is including 500,000 songs in its lyrics catalog, with more to come over time. Although Microsoft didn't announce this, the company's lyrics appear to be supplied by LyricFind, a lyrics service that has licensing deals with all four major music labels