I'm almost positive that the three heads are Dany, Jon, and Tyrion. They "fit" to me, in terms of what they offer. The thing about prophecy in this series is that it almost always has it somewhat wrong, it isn't really a guarantee that everything is going to be alright or all wrong, and several people from several parts of the world and religions seem to have similar prophecies. Take The Prince that Was Promised vs. Azhor Azzhai reborn. They are both very similar. We know that Azhor Azzhai is definitely not Stannis. The closest I came to believing in Azhor Azzhai was Beric Dondarrion, who was "reborn in fire" so to speak from Thoros of Myr. His blood was able to light his sword on fire.
With Jon potentially being killed, it seems logical to me that he would die, and Melisandre likely possesses the ability to resurrect him with a kiss. Of course, the kiss could also travel from Catelyn to The Wall, but Melisandre discovering this ability seems more likely, considering she is the most powerful of the red priests. I doubt the skill is reserved for the likes of Thoros of Myr, who didn't take his studies seriously yet still possesses the power of resurrection. He would have to be resurrected quickly in order to ensure that he would remain who he was, unlike Dondarrian who lost a shit ton of his memories or Cat who isn't remotely herself anymore, although the extreme cold to come can also play a large role in preserving his body and brain tissue.
Jon is the soldier of the group, the literal sword in the darkness, to steal a phrase from the Night's Watch, and I believe he would be the equivalent of Azhor Azzhai (but not the Prince that was Promised) reborn. He is also likely a Targaryan, however I do not believe that being a Targaryan is necessary for all three heads. Keep in mind that many signs are telling Jon that his place is at the Wall. He will never sit the Iron Throne, all of this prophecy is important because of the coming battle with the White Walkers, who will come out in a force never before seen, during a hellish winter. The battle between humans and whoever mans the undead is more important than the Game of Thrones.
Which brings me to Tyrion, who is perhaps the most intelligent person in the realm. He would serve as the braintrust for the three (Hand of the King/Queen), and only in his extraordinary intelligence do I believe he is one of the heads. The fact that he shows up in flames (in Melisandre's chapter, he was lot 97, which confused the fuck out of her) and Moquorro sees big things in the future for him, based on conversations, although I also believe Moquorro to be deceiving Victarion (Moquorro sees him being burned, and delivering the horn to Dany, I'm guessing).
And Dany is the Prince that was Promised. Nobody actually knows the prophecy, but if old Aemon Targaryan was certain of it, I believe it to be true. She is immune to fire, to the point where dragonfire only gives her a mild rash. In Dance with Dragons she fucks with her own prophecy a bit, showing us that prophecy is far from certainty, but she is the Prince(ss) that was Promised for sure.
Of the three, I know Tyrion has the weakest claim, but they do mesh perfectly. They have also been major characters for all of the books (despite their absence in Feast for Crows, but the only reason for that is that publishers are not fans of 2,000 page novels). Even so, based on what I've read I am absolutely certain that these are the three central characters.
I'm also really interested in the House of Black and White's interest in all of this, with the faceless man posing as Pate. It's important to remember that the House of Black and White is not atheist, and that the Many-Faced God is not a concrete figure to them. They believe in all religion. It makes sense that they would want a place in a prophecy that is showing up all around the world, mutated into several different forms and certainly corrupted, but almost omnipresent in the major religions. Arya might do some kick ass shit when the time comes, and while they knew about Needle, she never said they threw it away, as they might take the signs seriously, i.e. that she found Needle right before she found her way to Saltpans, where she met up with a ship that was conveniently headed to Braavos and knew what to do with anybody showing up with an Iron Coin yelling Valar Morhullis. If they do believe in all Gods, then they must know that the Gods have a plan for Arya of House Stark, and that they should not disregard it. The guy who sent her on her mission who might have been Jaquen H'gar (sp?) at one point (IMO) seemed to know she was a Warg as well. What happens with the Faceless Men excites the shit out of me, but they will play a role in the realm despite their pretended indifference to everything except mundane facts and prayers of death. Arya will be key to that.
And yes, I believe that one can be two or more of these things at the same time. The Prince That Was Promised is almost definitely one of the heads of the dragon, and it could have also been Azhor Azzhai and all the other things. I believe the three of these prophecies are from three different faiths, correct me if I am wrong concerning the Prince (ancient Valaryan), the Dragon Heads (many different faiths, including ancient Valaryan and The Undying of Qarth, as well as whatever the fuck Quithe practices), and Azhor Azzhai (red priests). There is a reason Melisandre became unsure and decided to remain with Jon at the Wall instead of joining Stannis. It may doom Stannis, and she still believes in him as of her last chapter, but the flames keep telling her "Jon Snow," so she feels fearful of leaving him. She is afraid that she is wrong about Stannis. She is so powerful it has turned her nasty and self-righteous, she might have trouble admitting that she misread everything with Stannis, even to herself. The trouble with the Red Priests is that they are zeolots, and they believe that the force they are messing with belongs to R'hallor, while it is really just a combination of myth and powerful magic. Stumbling upon such powers made them believe their own bullshit.