Over the years the rebbes and their followers have composed and sung hundreds of melodies. But, as happens with folklore, the melodies were sung but not written down (none of the rebbes had musical training). All that is changing now. There is a group (site only in Hebrew) which is recording the melodies and archiving them. They are collecting tapes and interviewing chassidim and have published two disks already.
Both David and I like this kind of music. A few weeks ago we attended a concert in Tel Aviv by the Israel Philharmonic Orchestra, the Yuval Cantors Choir and five guest cantors in a program of chassidic music. It was amazing! The clip below is of a Modzitz melody. (recorded with my still camera acting as a video so the quality leaves much to be desired but you get the idea)
Now, listen to real chassidim singing the same melody celebrating Tu B'shvat. (The melody starts around 5 minutes into the video right after the l'chaim.)