According to R. David de Sola Pool, "The Kaddish is in origin a closing doxology to an Aggadic discourse". It was not originally said by mourners, but rather by the rabbis upon the conclusion of a sermon or study session. It is written in Aramaic, which, at the time of its original composition, was the lingua franca of the Jewish people. The modern association between kaddish and mourning is misplaced, and in fact the version of kaddish colloquially known as "the mourner's' kaddish" is in the S&P custom chanted by the entire congregation on holidays, which is probably more authentic than it's recital by mourners only.
Unique to the S&P is their traditional pronunciation of three Aramaic words that occur in Kaddish:
Possibly the S&P actually have a tradition that this is the correct way to pronounce these Aramaic words; but it is also possible that they simply invented it as a way of distinguishing them from Hebrew words that otherwise sound identical.
Here are some traditional tunes for "Half Kaddish", the version said before Barechu:
♫ Weekday Kaddish [JC]
This version is also used for less important kaddishes in the Shabbat and festival services.
♫ Shabbat Standard Kaddish [JC]
♫ Shabbat Special Kaddish (first half) (e.g.before or on Rosh Hodesh) [JC]
I call this the Bach Kaddish because it recalls to me a famous and well-loved J.S. Bach melody.
♫ Shabbat Special Kaddish (second half) (e.g.before or on Rosh Hodesh) [JC]
This is the longer version of Half Kaddish sung on the High Holidays only:
♫ New Year Kaddish [JC]
Here is Kaddish "Yehe Shelama" - known in other communities as "Mourners' Kaddish", sung in a totally non-mournful way by the whole congregation:
...to be continued