If you don't know me by now (Simply Red)
You can get it if you really want (Jimmy Cliff)
If I had a hammer (Trini López)
If I were a bell (from Guys and Dolls, Doris Day version suggested by María)
If I were a rich man, a classical from Fiddler On The Roof. In this song you will also find some first type and mixed type (second and third) conditional sentences.
More than words, by Extreme (one of my favourite songs). Although the version below is clearer, this group sang this song live in Seville at the Expo 92. If you were lucky enough to see them (I wasn't), here is a reminder: http://www.youtube.com/watch?v=yE1XRob6KkU
In the following song, the subjunctive "if I were..." is replaced by the more and more frequently used simple past "if I was...", something considered incorrect from a traditional grammar perspective:
All I want is you (
I also think "If I could" by Seal is a good song for this purpose but I couldn't find a listenable version on the web. Any more suggestions?