Dumbledore says that Voldemort transferred some of his powers to Harry, the mechanism for which is later revealed to be the soul fragment lodged in Harry’s head. But the only power he ever singles out is Parseltongue. Were there others?
Well, what is Voldemort good at, and does any of it correspond with what Harry is good at?
We never see Voldemort cast a Patronus. That one might be all Harry.
Voldemort is good at casting the Cruciatus Curse, obviously. Bellatrix taunts Harry for only causing her a little bit of pain on his very first ever try, which could be taken to mean Harry doesn’t have a natural talent for it… but then again, we don’t know what’s normal for people trying to learn the curse. Maybe causing a bit of pain the first time and performing perfectly two years later is actually pretty good. How would we know? It’s not like the DA practiced it so we could see how Seamus or Parvati would do on their first try. Harry’s Cruciatus aptitude could fall anywhere on the spectrum from poor to precocious.
Harry succeeds on his first try at the Imperius Curse, though he has to renew it a few times, which suggests he hasn’t fully mastered it. But again, we don’t know how another kid his age trying it for the first time would fare. Draco is good at it-but he probably had some training from his family before he cast it on Rosmerta. Auntie Bellatrix taught him Occlumency; why not the Unforgivable Curses too? Lucius also knows Imperius, and surely that would be a good curse to teach your son for his future political career. (As long as you don’t have any scruples, that is.) McGonagall is also successful, but she’s had decades more to practice magic in general than Harry and is noted as a powerful witch and a forceful personality, which probably counts for something. And we don’t know that she’s never cast it before. So Harry’s performance probably falls somewhere between okay and very good.
There are other hints we can use to narrow down Harry’s talent at the Unforgivables. Bellatrix tells Harry that the Unforgivable Curses in general are difficult in that you “have to mean them.” This may mean having the right emotional state, as she claims, or it might be a matter of strength of will, and her interpretation suits her personality. Is there any other support for the idea that they’re especially difficult somehow, not just rarely performed because they’re illegal?
I think so. Barty Jr. (as Moody) says that the Killing Curse takes “a powerful bit of magic” and that the class could all cast it on him at once without giving him so much as a nosebleed. If he’s exaggerating, it probably isn’t by much-his statements have to sound convincing if Dumbledore and the other staff learn about them. And identity theft aside, Barty is fairly truthful.
We also see combat situations where the DEs mostly don’t rely on the Killing Curse, even when it would be useful. Why not AK one of the kids in the Ministry battle to show you’re serious and then tell Harry to hand over the prophecy, for instance? And in the Seven Potters Battle, when they’re throwing Killing Curses around like confetti, they manage to kill… Hedwig. (Voldemort kills Moody, but we already know he’s good at the Killing Curse.) Did the the Death Eaters just have terrible aim, or were they throwing around Killing Curses too weak to kill humans?
So yes, the Killing Curse probably really is hard. That supports a third of Bella’s claim, and makes it more likely that she’s telling the truth and the other two are also hard. In which case I’d upgrade Harry’s performance to “probably pretty good.”
Okay, what else? Voldemort is an extremely powerful Legilimens. Harry never tries to learn that skill. However, in one Occlumency lesson, he breaks into Snape’s memories. Casting a Shield Charm might help push an intruder out of your head, but it’s hard to see how it would carry you along into theirs. And while Snape might not have his full Occlumency defenses up, he probably isn’t leaving his mind completely undefended; he knows Voldemort could be peeking out of Harry’s eyes. Snape is a superb Occlumens, so light Occlumency defenses for him are probably formidable. Yet Harry gets through. Without meaning to, or even realizing he’s doing it-much as he speaks Parseltongue without realizing it.
Similarly, while Voldemort figures out that he can actively peek into Harry’s mind from a distance and send him false images, Harry never gets that kind of control over the connection. But he is remarkably good at accidentally peeking into Voldemort’s mind, even when it involves sliding another link down the chain into Nagini’s mind. Many of these viewings happen while Harry is asleep, while his conscious Harry-mind is offline and the Voldie-bit might have more control. So maybe it isn’t that Harry is good at using the Voldiecam, exactly-the Voldie-bit is. All of this suggests that Legilimency could be another “transferred” power which Harry can only access imperfectly.
Finally, Voldemort is good at wandless magic. We don’t know whether Harry has more or more dramatic magical outbursts as a child than usual. We don’t hear of any other Hogwarts students wandlessly blowing up their aunts at age 13. Did it just never come up, and this sort of thing happens all the time to Muggle-raised kids who can’t use magic during holidays? Or is Harry especially good at it? (Power-wise, not control-wise.) Ariana fatally blew up her mother and herself wandlessly, but Dumbledore’s sister might have more raw magical power than the average witch. The few characters who perform wandless magic on purpose (Dumbledore, Voldemort, Snape, and… anyone else?) are flagged as unusually talented in general. Harry might be unusually powerful too, but his wandless outbursts might also be a Voldie-power.
(ETA: How could I forget young Lily deliberately, wandlessly floating off swings and opening flowers! So Harry might have inherited that talent, if it is heritable, and just isn't as good at controlling it as Lily or never made the effort to become so. But that doesn't preclude it being reinforced by the Voldie-bit, so we can have both.)
I wonder whether Harry’s Amazing Auto-Wand is actually not about his wand, but about him once again using a Voldie-power imperfectly. Instead of just making light appear, he lights up his wand from several feet away (OotP), and instead of blasting Voldemort entirely wandlessly, he makes his wand shoot golden fire for him, apparently without knowing any such spell (DH).
So, while it isn’t conclusive, I think it’s suggested that a lot of Harry’s more impressive magic-especially that which he performs without being fully conscious that he’s doing it-is leakage from the Voldie-bit. Maybe now that it’s gone, he wouldn’t be nearly as successful at casting the Cruciatus and Imperius Curses or peeking into anyone’s mind by any method, and his wand will never again act “by itself.”