At both of my independent minyanim, we use a wide variety of tunes for "Adon Olam", some "traditional" and some rather untraditional (like using the tune for "Down in the Muddy, Muddy" for the Shabbat when we read Parashat Noach). Here is a story with a link to a wonderful a capella rendition of Adon Olam to a modern tune: Adon Olam to Tune of Pharrell's "Happy" I'm not with-it enough to recognize the tune, but I really liked it. Definitely too fast and complex for my either of my minyanim though.
Here is a medley of many different Adon Olam tunes, many of which my minyanim has sung: Adon Olam medley A couple of years ago, my family was given the honor of leading this hymn at the bat mitzvah of the daughter of our college friend. Unfortunately, we were told that we had to use the tune that I like least of all which is the first one of the above medley. I just think it sounds too dirge-like.
My daughter spent a summer in the Hebrew immersion program at OSRUI, a camp of the Reform movement. She thought it was really weird that they always sang Adon Olam to the same tune which composed by a famous Israeli singer, Uzi Chitman: Adon Olam - melody by Uzi Chitman