I'd link to a Mary Sue Litmus Test, but considering the sheer variety of these out there, I doubt that any one of them is truly representative of all the rest. Considering how much they've been copied-and-pasted, I suspect that while there may be an original (for the Gargoyles fandom, I believe) and several generic ones, they all strive for some strange Platonic ideal that can be viewed, however dimly, only by taking these in aggregate. Some questions are fandom-specific, and some are not. Even those that appear out of place may need to be asked anyway, to account for the stories where the main character is a fan of some contemporary pop or rock group while simultaneously being a native of some fantastic time, be it in the far future or distant past.
I should probably mention here that I don't actually have anything visceral against the Mary Sue. I believe that all the accusations thereof can be boiled down to simple Bad Writing, which is too general a thing to be confined to a single cause. Besides, I used to be a member of the Mary Sue Appreciation Society (RIP Kielle), primarily for the reason that writing, or at least amateur writing, should be fun, and if a Sue is involved, so be it.
Anyway. The main reason I brought up the Litmus Test is that their degree of usefulness can be determined by reversing the purpose, and testing the tests themselves. The example character to put through the questions is not some canon or well-accepted character, but yourself. As in, your Real Life self.
If you've existed for a sufficient period of time (late teens or so) and had a somewhat varied life, chances are you'll score at least in the "Danger" section, and very likely in the "Outright Mary Sue" category. The (un)luckier of us might even reach "Uber-Sue", the highest goal that may be achieved. The points come from surprising places: for example, if I discount my scattershot knowledge of Japanese (as well as my efforts at Klingon), I am still relatively fluent in three languages (English, Mandarin Chinese, Bahasa Indonesia), which is apparently a Sue trait, despite also being part of my cultural heritage. In fact, simply being Asian (the tests seldom distinguish between East Asian or South Asian or whatever) is worth Sue points. Considering the overall population of the world, this has always struck me as odd.
On an intriguing tangent, anime fanfiction fandom in general reverses this: in a standard Japanese setting, a "foreign" character (often labelled "gaijin", or "outside person"/"outsider", which is one of the many Japanese terms made popular in Anglophone communities by anime fandom, despite the more appropriate "gaikokujin", or "person from another country") has the same connotations as "Asian" characters in more Western fandoms. Exotic, different, special… basically a "look at me!" attention-grabber, which is, at base, the point of a Mary Sue.
I've always held that the Litmus Tests are originally meant to be jokes. Unfortunately, I've met both creators and takers of the tests who treat them with the utmost seriousness, so apparently I am mistaken.
The point of all of the above ramblings is that as an audience, we tend to have strange definitions of "realism" when it comes to characters. I submit that we do not, as such, want "realistic characters". Instead, we want simplified versions of what we believe to be realistic characters. (Yes, I know about exceptions. I'll get to them in a much later post.)
A passage in The Science of Discworld II alludes to the idea that while we place our own freedom of will pretty importantly, socially we don't expect anyone else to have any. This can be seen in the phrases "not himself" and "out of character", which we often apply when a character is acting in an unexpected manner, and our reactions and expectations of the reasons behind those actions. We're happiest, the book notes, when the explanation turns out to remove the element of free will from the situation (doing it under duress, or doing it for a bet, or whatever). Of course, the question I'm interested in is not that we don't expect others to have free will, but whether we have free will in the first place, especially if we do have it, but our choices are constrained by what other expect us to act like, or just our own mental and moral boundaries.
Therefore, an actual "real" person would be far too complicated to present to the viewer. This is partly a limitation of the sense of narrative which a work of fiction has to establish: yes, I can speak three languages, but that is not always relevant to whatever story I might self-insert myself in. A viewer expects not to see a person, but a bundle of character traits mushed together, the same way we see everyone else around us.
And the average viewer will probably have some character traits they like, and some they don't. Thus, the more "complex" (ie the more character traits they have) a character, the more likely that one of these undesirable traits will sneak inside. Sometimes these are excusable, or easily overlooked in favour of the more welcome traits, and so they get filed away under "bad, but trivial, habits", or some such. After all, we do the same in Real Life every time we meet someone new.
A "realistic" character is thus someone who presents to us the right number and type of character traits, that we may assign them some convenient mental labels in the same way we do in Real Life. The difference is that the Real Life version tends to have more, while the character can be a mere hollow shell in comparison. There is, after all, no need for further expansion of the character, without going into irrelevancies: we know quite a lot about Nanoha's character, for example, but we don't know much about her opinions (if any) on the Hanshin Tigers. Mind you, our beliefs intertwine and affect each other to a large degree, so Nanoha's views on, say, the global economy may be influenced by her general political views, and knowing those would help us write her "in character", even in non-political situations.
So take a bundle of character traits, and make that bundle bigger. Add in more traits, more aspects, more personality parts… and you'll find yourself ticking off plenty of Mary Sue traits in a Litmus Test. Whereas if you have just a few character traits in that bundle, the character is accused of being "bland" and "cookie-cutter". Their bundle-ness becomes more pronounced and obvious, rather than hidden behind the mask of being a character.
This isn't a one-dimensional sliding scale, or any scale of any sort. The Mary Sue is unwelcome not so much because they can do everything (to exaggerate the position), but because they do everything, or at least the major stuff. It's possible to have an Uber-Sue according to the Litmus Tests and have a bland, cookie-cutter character who doesn't register on the narrative at all; if the character doesn't get a chance to exhibit all that competence, there's no spotlight to hog.
It's an interesting balance, different for every character and every viewer. This is why some characters can seem like two-dimensional cardboard cut-outs to some fans, and also well-defined, realistic epitomes of what a character should be designed as to others. If all else fails, there is always the equally human tendency to project our desires and hopes onto a relatively blank slate, writing narratives in our heads for what an ill-described character should be like. Frequently-encountered tropes help with this: recall Konata's insistence that the twintail hairstyle is a common feature of tsundere characters, and vice versa.
Or, of course, I could just be talking out of my arse.