I am a firm believer in The Muse. This is not an uncommon belief. In artistic or poetic experience, and this can be as true of a mediocre artist or poet as any other, one sometimes has a sense of the work as not being wholly dependent on oneself, but as nonetheless coming together in a way that is not merely aleatory, as if the work were in some way flowing through one from something else -- as being inspired. This something else, whatever it may be, is The Muse.
The Muse has a fair degree of support from experience. One can experience it oneself, although it takes a certain amount of immersion in an art, and can only be had sporadically. The experience itself is not a perception of The Muse, but rather of oneself as effectuated ('inspired') in a work by something other than oneself; it is a sense of what we might call relatively dependent receptive cooperation.
(1) It comes across as a feeling of working with something. All of art, of course, is in some sense a 'working with' something, a cooperation with materials in particular. But the feeling associated with The Muse is not a cooperation with materials; indeed, it is very distinguishable from that. It is a feeling of receiving something in one's work from somewhere else. And this is not the sort of thing that would ordinarily mark out an experience of luckily hitting on the right way to solve a problem; it is a sense of something very orderly. Thus it does not appear to originate wholly within oneself, and its working is too orderly to seem like chance
(2) It comes across as a feeling of relative dependence. What one receives seems to be higher than one's skill, and (again) not in the way of luck, but something much more structured, more ordered.
This experience is quite common. There are many different accounts of what The Muse is, but you can talk to artists and artisans and poets from many different cultures and you can find across very different cultures those who recognize immediately what you are talking about. Thus The Muse is supported by something like a consensus sapientium; that is to say, those who know art tend to know The Muse. Conceivably there are those who could be great artists or poets without ever having the experience. But it seems a widely shared human experience among those who are at least competent in, and devoted to, their craft. What is more, belief in The Muse, in some form, is spread much more widely than just among those who experience it; taste is one with genius in suggesting at least something like it. Aesthetic experience has a sort of related experience; before great art, or reading great poetry, people sometimes have the experience that this goes beyond mere skill. It is harder, I think, for aesthetic experience to distinguish inspiration from good luck. But people who love art or literature, even if they know nothing about producing it, do sometimes have the experience of something so excellent, so breath-takingly right, that either it must have been inspired or the artist must have been astoundingly lucky in hitting on exactly the right thing to do. And if we needed further confirmation, we could perhaps go to the close association through history of religious experience and artistic experience. Indeed, of course, calling on a god or goddess to inspire one is a common literary trope, arguably (depending on how one translates and interprets certain expressions) going all the way back to the first poet whose works are extant and whom we know by name, Enheduanna, recording her songs to the goddess Inanna.
So let it be granted that it is an actual experience. Is there any particular reason to think it illusory? We have to be careful about what this means; 'The Muse' here is just a generic description for the something else moving one to greater work than skill can achieve, and the experience is actually of oneself as being so moved, not a direct experience of that something else. So for it to be illusory would have to mean that, in fact, either the experience had no connection with the art (so it is epiphenomenal mirage) or the artist is experiencing something that does not actually go beyond himself (so it is a misinterpreted experience of one's own imagination at work) or the work goes beyond skill but by mere luck and this experience is a 'filling-in' of a gap (so it is a pareidolia in which the mind treats chance as if it were more like design or order). The first of these alternatives seems to be entirely inconsistent with all evidence, and it's difficult in any case to make sense of a purely illusory experience that we have for no obvious reason, and why we would simply dismiss apparent evidence wholesale. So misinterpreted self or luck seem the two most straightforward alternatives to The Muse.
A. G. Tyson had a (not particularly good) book published in 1853 called, An Essay on the Poetical and Musical Customs of the Ancients, in which he has a section attacking the notion of poetic inspiration. He doesn't actually give much in the way of argument as opposed to abuse, but he says of Aeschylus that "his supposed inspiration was no doubt the force of his strong and fervid imagination, fixed on a particular purpose" (p. 119). This is a misinterpreted self view. Put in this form, it is not, I think, even a remotely plausible interpretation; it is part and parcel of the experience of poetic inspiration that it is of something beyond things like "strong and fervid imagination". But you could have views that would be more tenable -- for instance, if you attributed it to subconscious or unconscious processes rather than conscious ones. (The primary difficulty there would be keeping it distinguished from The Muse; the less conscious it is, the more it starts sounding like The Muse.) Luck runs into the problem of orderliness noted above, as well as the widespreadness of the experience and sometimes a consistency across sporadic experiences, but a version of it might still be viable if you held (for instance) that artists and poets are ceaselessly trying so many different things that they will inevitably hit by luck on some extraordinarily useful solutions, even if the latter are very unlikely. In any case, whether misinterpreted self or misinterpreted luck, I don't think the experience itself can definitively refute them; the problem, though, is that both require a good deal of independent hypothesis, and neither really gives much of an explanation for the feeling of causality involved.
There is a view that one could take that is in a sense midway between these and more substantive accounts of The Muse, namely, that the sense of poetic inspiration is actually a sense of self-organizing formal elements in the artistic activity. This gets the feeling of relative dependence right, and it lets one credit this as a genuine experience of something that is in some sense not oneself. But the experience is taken to be one of a kind of necessity, a sense of things having to fit together in a certain way. The primary difficulty with this is that when you look at what people attribute to The Muse, the actual alleged contribution doesn't usually seem to have this sort of necessity -- it often seems quite contingent and unpredictable and especially difficult to fit to any particular rule -- and the arts are so very different that it seems odd that we would have the same kind of experience across arts with such very different formal elements. Some things that are apparently quite plausible for one art -- e.g., language as a sort of collective unconscious (Schaeffer) -- are not necessarily so for another.
If we do accept The Muse, then as I said this is here a generic label, and there are many different things it could be, and many different things have been put into the slot: God, Forms, gods, angels, collective unconscious, world soul, or what have you. Nothing strictly requires that it be one and only one thing; you could be a pluralist about The Muse. The big divide seems to be whether The Muse is considered personal or impersonal. Tyson gives a brief argument against the view that poetic inspiration could be from anything divine ("Supposing that we had a godly inspiration the productions thereof must of a certainty be godly -- be of one uniform and specific character"), but it is not a very good argument. If we take a poem, for instance, The Muse does not write the poem; the poet does. This is integral to the actual artistic experience. The Muse is not like dictation or automatic writing. But the poet experiences the writing of the poem as something he or she does in trying to keep up with something 'flowing through' him or her -- being carried along by and trying to keep a hold of something beyond his or her skill, often very shakily. The Muse isn't usually thought of as simply handing you the poem; it moves you to make it, and the movement is itself something different from your ordinary movement. Poets very often have a sense have a sense of stumbling over their own weaknesses and limits in trying to keep up, just as you might stumble while trying to keep from falling out of control if in walking you were being pushed along in a certain direction by a force that you did not control. Whether one takes The Muse to be personal or impersonal, I suspect, will also really depend on one's broader metaphysics rather than anything that can be found in the experience itself, just as the bare experience of being pushed along is not itself really different regardless of whether the thing pushing is a current or a hand.
So from the experience itself we get a very limited conclusion: sometimes the artist is caused to do more (inspired) by something other than himself or herself, in the very activity of artistic production. The most natural reading of the experience is that this is due to an actual cause distinguishable from oneself or mere luck. Or, to put it in other words, we sometimes have an actual experience of being inspired by something beyond ourselves; the most natural inference, barring strong argument in favor of some alternative, is that this is due to at least some actual inspiring cause or causes of some sort. This is defeasible, of course, but the actual alternatives that have been proposed have tended to be quite handwavey and arbitrary. Or, to put it a third way, artistic inspiration is real, and really inspiration, whatever the precise explanation for it might be.