I opted not to write a whole post on this topic. I wanted to. I thought that I might but I am lacking the passion. So instead you can go to Treppenwitz and see what prompted my thought.
One more comment about this. In my experience Havdalah and Adon Olam were the most frequent targets of being sung to an alternative tune, such as the theme to Cheers, Oklahoma or Back In The Saddle Again.