There is very little on the net with regards to the origin of non monotonous singing. Can any one point me to a decent reference?
Question: What religions besides Christianity employ singing as a part of their worship?
Let’s try again.
Do Muslims, Jews, Hindus, Silhs, Wiccans or any other non Christian religion employ vocal melody in their worship ?
Well, There are specialists in Jewish religious music called ‘cantors’. So there’s a yes for you.
So called because the first one was berated by the Rabbis to quit ‘singing his prayers’.
He responded “I cantor God will be displeased with me.” 
Um, unless you have some kind of ban against religion in your worship, every religion does, I would think so. I can answer for Buddhism, yes. Hinduism, absolutely, especially in North India. (I can answer better for North India is what I mean). Islam has a beautiful custom where they will sing as though they are the female and God is their lover* and God woos them so beautifully away from the real world.
Islam:
“Chaap tilak sab cheen re
mose naine milaike”
“He took my jewelry and my decorations,
When his eyes met mine”
It is sung like a beautiful love song but it really is referring to God, who took all of their love of material possessions and turned it into the Divine Love of God.
Hindus sing to Krishna, who danced and played with his gopis, often about “don’t tease me”, “don’t break my pot” as he was very mischeivous.
Sikhs have a reference to God in many, many of their folks songs. Almost everyone will end with a line something like “And thank God for this beautiful spring/holiday/baby/marriage”.
Other people can answer about the others.
It’s very beautiful, and one of the things I refuse to lose my love for even though I am atheist is religious songs. They are usually sung with such feeling and such heart, you can’t help but love them!
*Lover very much in the old-world sense, where your lover wooed you and romanced you, not sexually of course.
I know of no religious group that omits all song from their liturgies.
The position of cantor is an important role in Jewish worship.
There are several strains in Islam that choose to eschew or shun music, but to do so, they simply define several chanted prayers as recitation rather than song. The Sufi tradition includes a lot of musical forms, although they tend to be performances intended to inspire rather than chants in which congregations participate. (Given the distinctions, I suppose it is possible to say that (many) Muslims have no songs related to religion. It probably is accurate to say that there are no liturgical hymns associated with Islam, although I am open to correction.
Hymns play important roles in both Hindu and the several strains of Buddhist worship.
As far as I know, the Sikhs (I presume we were not talking about the Sith), following on their Hindu relationships, also include song in their worship, although I believe that, with their Muslim connections, much of that music is in the form of inspirational performance rather than congregational participation.
I’d say a typical Jewish synagogue service is at least 90% singing or chanting. Sometimes it’s the cantor that sings, sometimes it’s responsive singing, sometimes everyone sings together. Virtually ever prayer has at least one tune.
Furthermore, if you look at a Hebrew bible, you’ll see little lines and squiglles over and under the word. Those are musical notations - the entire Tanach is meant to be sung.
So yeah, singing has a very inmortant role in Judaism.