What are the theories regarding the origins of Qur'an ?

There are probably as many theories regarding the origins of the Qur'an as there are western scholars of the Qur'an.

In general terms I think Western academics are actually quite sympathetic to the notion of revelation to the prophet Muhammad, although not perhaps in a way that would be religiously orthodox to devout Muslims, insofar as many if not most Western academics are willing to at least entertain the idea of the Qur'an as the product of a religious experience genuinely felt, regardless of where they come down on Muhammad's actual prophethood.

That being said, and I give this as an example rather than as any kind of systematic theory for Qur'anic origin options, it might be helpful to, among other options, divide theories of the origins of the Qur'an based on their dating of specific verses, and of the size of the unit of revelation. A third category, although one the historiography of which I am much less familiar with, is the extent to which the Qur'an owes its content to previous religious scriptures or documents.

Obviously the dating of the Qur'an is an important question for its origins. In part the debate over this emerged from a debate within Islam about the order in which individual Suras were revealed. While there is a "standard" 21st century order that most believing Muslims refer to and which separates Suras as "Meccan" or "Medinan", this "Cairo" or "Egyptian" ordering was actually only accepted as standard in the past couple centuries (within the early 20th, I believe).

Among Western academics the major modern work on this was done by Theodor Noldeke, who wanted to not only date and order the Suras, but also to periodize them (Early, Middle and Late Meccan, and Medinan), as well as to describe certain characteristics of these respective periods (so as a very broad example, early Suras refer to God as "ar-Rab" "The Lord" while later Suras refer to God as "Allah" "The (One) God".

Noldeke's mission to carry out this kind of philological analysis was continued by others, but for my purposes here it's Richard Bell who stands out. For Bell, trying to date and order individual Suras was, I won't say impossible, but far less straightforward than one-verse one-period, because the individual verses, to his mind, were hopelessly out of order and the surahs were filled with later glosses and insertions. He looked for things like unusual line insertions that broke up otherwise rhyming portions of text, or blocks of text that seem incoherent with what comes before or after. Obviously Muslims recognize that longer surahs were not revealed all at once, but for the purposes of theories of the Qur'an's origins Bell posited that these units of revelation were in general very small and made all the more complicated for purposes of dating by these glosses and insertions. His Introduction to the Qur'an detailed these theories which he then tried to assemble together in The Qur'an: Translated, with a critical re-arrangement of the Surahs.

More recent scholars have pushed back against this, looking at some of the interesting ways in which even the longer surahs of the Qur'an can be viewed as much larger totalities. So where Bell sees a very narrow fracturing, someone like Angelika Neuwirth or Michel Cuypers look at features like ring structure in the longer surahs to point to a coherent wholes.

In terms of dating of surahs, this means that for someone like Bell dating the Quran is almost an endless task in which the Quran can be broken down into infinitely smaller bits to be analysed and dated, while for the latter group it's a much more plausible enterprise.

While the above is perhaps a kind of literary analysis of the Qur'an's content and structure there are of course also physical manuscript analyses to be done. One longstanding theory of the Qur'an was that it may have originated or developed long after it was supposedly revealed to the prophet, perhaps over a period of time before settling into its now-known final form (basically by analogy to the Christian New Testament). This, to my knowledge, has been mostly refuted by the manuscript evidence that we have, which show a very low degree of variance in Qur'anic manuscripts which have been dated very early, for instance, the Sana'a manuscript. These differences, while perhaps troubling to the devout, are pretty "normal" for historians. So in that sense the general consensus is that the Qur'an is from the time it purports to be and is, in all likelihood something that was genuinely revealed to Muhammad (as he understood it) and not, say, a later forgery or something like that. Even Patricia Crone, who is perhaps the most well-known of the Islamic source skeptics, agreed that the Qur'an (along with the constitution of Medina) is in all likelihood a 7th century document composed by Muhammad. That being said there are still fringe revisionists who posit that it was all a later construction and that Muhammad might not have been real, etc. But as I've said this is not the mainstream view.

That being said, and as I said at the top, there is also quite a lot of discussion about the extent to which the Qur'an may have borrowed from or been based on earlier texts. Here I am not at all familiar with the scholarship, except to say that I don't think anyone is suggesting that the Qur'an is cut wholecloth from such texts, but rather is arguing about what texts or contexts Muhammad might have had access to or been dealing with in the Arabian religious milieu of the 6th/7th century AD.

For an overview of much of this historiography as well as quite a good look at the kind of western academic analysis of the content of the Qur'an I would have a look at Neal Robinson's Discovering the Qur'an which is even-handed, comprehensive and well-written.