I, personally, prefer full-length album versions over shorter single versions. I kind of feel like I'm being cheated when I hear the shorter version.
I agree 100% when it comes to "Night Moves" by Bob Seger. WMYI (My 102.5) here in the Greenville-Spartanburg area plays the short, chopped-up version. Even more annoying, the internet radio station Mellow Gold (on Shoutcast) also plays the short version! I can understand why a corporate owned FM plays this version (cram in more commercials, of course), but a jockless, commercial free internet-only station? On the other hand, ABC's (or is it Citadel now) satellite oldies format (heard here on WQUL 1510/95.9 out of Woodruff) plays the more familiar, full-length album version.
On the subject of Billy Joel songs, "Just the Way You Are" is another one. The shorter version removes the entire second verse; this is the version most commonly heard on the radio. The Greatest Hits CD, of course, has the short version (although the liner notes contain the lyrics to the full song!)
"Lady" by Little River Band hasn't been mentioned yet. The short version removes the guitar riff from the beginning, as well as the entire second verse, then fades out halfway through the chorus, right about at "I think it's only fair I should say to you..."
"Three Times a Lady" by the Commodores. This one also has an entire second verse that is removed from the more commonly heard single edit version.
Another one that surprised me was "Lowdown" by Boz Scaggs. I never knew there was a shorter version of this one until I heard it on Sirius's 70s channel (which was playing out of the intercom at my previous job); the song was sped up and the second verse was removed! (Surprisingly, they later switched to playing Sirius's "Blend" adult contemporary channel out of the intercom, and that channel played the full-length album version of "Lowdown." The Blend has since removed 70s music from their playlist, so "Lowdown" isn't heard there at all anymore.)
Seconded about "With a Little Luck" by Paul McCartney and Wings. The only copy of that song I had was on a cassette tape of All the Best, which, of course, has the chopped-up version. I have since downloaded the album version from iTunes.
"Wildfire" by Michael Martin Murphey is another one. The album version has a piano solo at the beginning and end, both of which are removed on the single edit version. Surprisingly, the now-defunct Timeless soft AC/adult standards satellite format actually played the full-length album version of this one; satellite formats usually stick to the shorter versions of songs.
Regarding supporting songs- one you almost never hear is the one at the beginning of "Out of Touch" by Hall & Oates; on the album, there is a fast-paced, mostly instrumental intro called "Dance on Your Knees."
Another one is "Hard to Say I'm Sorry" by Chicago, which has a more upbeat song called "Get Away" at the end. I know WGFB (B-103) in Rockford, Illinois played this song with "Get Away" intact, but most stations I've heard it on fade the song out before that point.
Then there's "Tainted Love" by Soft Cell, which has a cover of the Diana Ross and the Supremes classic "Where Did Our Love Go?" at the end. Some stations play the entire double song, while others only play the first song. (On one CD copy I have of this song- Time Life's Modern Rock collection, the song is labeled as "Tainted Love/Where Did Our Love Go", but only has "Tainted Love" on it! Got ripped off there.)
When it comes to various artist compilations, it's pretty much a crapshoot as to which version of a song you're going to get. Seems like most compilations contain the shorter version, although I've been pleasantly surprised to get the longer version on occasion. Greatest hits albums, more often than not, contain the shorter single versions. The only way to guarantee you'll get the album version is to pony up the money for the actual album.
When I operated my part 15 AM station, I made it a point to play the full-length album versions of songs. However, if the shorter single version was all I had, I reluctantly went ahead and played it, until I had the money to get a CD of the album, after which all subsequent plays would be of the album version.