I think the most interesting controversy at the moment is whether Young Griff/Aegon VI is actually the son of Rhaegar Targaryen, a Blackfyre pretender, or Magister Illyrio's son.
The most considerable evidence for him being the real thing is what Varys said to Kevan right at the end of A Dance With Dragons, when nobody else could hear and when Kevan was about to die. He described Aegon and said that he had received training worthy of a king and had landed in Westeros. This seems to confirm that Aegon is alive, and although it doesn't confirm that he is Young Griff, it seems unlikely that he could be anything else.
Jon Connington seems to believe it, but that's not much evidence of anything.
Daenerys has a vision of a "mummer's dragon", but this is ambiguous. It could either mean:
a) He isn't a real dragon at all, and therefore not a Targaryen
b) He is a dragon, and the mummer in question is Varys, who is certainly controlling him to some degree.
He also has the Targaryen features, but so do the Daynes and people from Lys, and I'm sure many others.
Because of the first bit, however, I still lean towards believing that he is the real Aegon.
(Former) Lead Moderator and (Eternal) VGC Detective 
