A caveat (of which I have many) before I begin: this is a list of books I've read where the protagonists have excellently glaring flaws. However, those flaws go hand in hand with the characters themselves; they cannot be divorced from one another. And just as we ought not try to put asunder what the author has joined together, as authors we should not try joining together what should stay asunder! We can't throw darts at a dartboard of character flaws in order to choose which ones our protagonist should have. These grow out of the person himself, and develop with him; they must be intrinsically a part of him.
There's my caveat. Now we can move on to fun stuff.
north and south
In talking of flawed characters, my mind flew immediately to Mr. Thornton of Elizabeth Gaskell's North and South. Not surprising, since he is one of my favorite characters ever. But anyhow, those of you who have either seen the film or read the novel will understand immediately how he represents my point. His flaws are obvious: pride, a sharp tongue and quick temper, and perhaps overmuch ambition. They reveal themselves in ways that hurt a number of people, especially the workers in his cotton mill, for they make him nigh oblivious to their suffering. He is no saint, and his flaws are no mere trifles; they have keen effects on those around him.
With flaws like those, he could easily become odious to the reader. Gaskell pulled it off, however, by balancing these elements of his personality with other, equally critical ones. He is a hard worker, glad to break his back in support of his family; he loves ardently; and he is not lacking in compassion, though he shows it harshly. He is certainly a conflicted personality, but it all comes together to create someone who is very real and very much a hero in his own way.
Another obvious choice! Who doesn't think of Holmes when flaws are mentioned? There are few elements of his personality that don't constitute flaws. He is arrogant, rude, selfish, oblivious, manipulative, verbally abusive (sometimes), and a drug-addict. He's not exactly the spitting image of a hero. And again, these things are not whitewashed - they're out in the open for all readers to see. We really ought to hate him. But most of us don't, and for some crazy reason he so endeared himself to readers that there were riots and protests when Conan Doyle attempted to kill him off. For he is also brilliant, witty, at times kindhearted, and even occasionally just plain wrong.
the chronicles of narnia
Of the Pevensie children, Edmund and Lucy are by far the most thoroughly developed and the best-loved. Edmund is a very flawed personality: he went and betrayed his siblings, after all, and was just an all-around brat who needed a good swat on the rear end. But we love his redemption, and even the natural roughness of his personality toward a character like Eustace Clarence Scrubb is attractive. (Because Eustace "almost deserved it.") Lucy is not as obviously flawed, but she still has her weaknesses - her jealousy of Susan, for instance, which pops up in The Voyage of the 'Dawn Treader.'
a tale of two cities
Um, Sydney Carton. Need I really say any more? Even more than Thornton, even more than Holmes, Carton represents an anti-hero. He's a drunkard and a ne'er-do-well, just the sort of Dickens character you are meant to loathe. But instead you pity him for being, it would appear, incapable of change - for being chained to his vices - for his unrequited love. And then you're blown away by the ending, sob over him, and love him for his nobility. End of story.
the count of monte cristo
Here you have a main character bent on revenge, obsessed with the idea of being sent by God to bring evildoers to justice, ruining people's lives left and right. He has so many flaws, there are very few bits of gem left in the whole lump. If you dig around a bit, though, you find that he is capable of some form of compassion toward those he considers innocent (does that even count?), and of immense generosity - no stinginess there! I am actually hard-pressed to think of anything else. Please call back at a later date.
The first flaw in the hero of Megan Whalen Turner's series is self-evident: he's a bit light-fingered. He also lies and swears, so you could call him light-tongued as well. He is horrendously proud, often sullen, frequently bitter toward both the gods and the people around him. Actually, he's very flawed indeed and makes the reader want to hit him upside the head. He's also in love, and it's unrequited - both things that tend to make the reader soft-hearted. In addition, he is incredibly loyal and at once brave and oddly fearful. He is a well-blended mishmash of traits, and one of my favorite things about The Thief and The Queen of Attolia.
howl's moving castle
I almost forgot this gem, and that would be a heinous crime. How can you leave Wizard Howl out of a mix like this? He is talented, but on the other hand, he's a coward and what another character calls a "slitherer-outer": he won't face any danger if he can help it. He's also quite heartless and has a habit of making girls fall in love with him, then leaving them in tears. But that's not his fault, now is it? And his wit (ever a popular trait), his humor, and his character development make him loveable despite these things.
For amusement's sake, I'll do a run-through of the most glaring flaws in all these characters. Pride; excessive ambition; arrogance; rudeness; selfishness; drug-addiction (!); manipulation; betrayal; jealousy; drunkenness; idleness; hypocrisy; hatred; thievery; lying; bitterness; swearing; cowardice; and heartlessness. Not the marks of heroes, we would think, and yet borne by heroes. They are the marks, or some of the marks, of fallen men and women - and that includes those who are saved and being saved, but who are not yet "confirmed in righteousness." There are still flaws that go down to the bone.