In Game of Thrones, the priests of the Red God and the Old Gods of the Forest exhibit magic powers. Do the devotees of the Seven exhibit similar power?
If they do, they are considerably more subtle powers than those of R'hllor and the Old Gods. During my re-reads of the A Song of Ice and Fire series I have seen no such overt examples of magic powers. It would seem most of their power is non-magical, in the sense that they have many (zealous) followers, martyrs and a strict code of conduct.
In A Feast for Crows there is a chapter when Brienne and her small party visited the Quiet Isle, where believers of the Seven go to atone for their sins. The Elder Brother there had an aura of mysticism about him, was said to be a great healer and did... (Mystery Spoiler)
...apparently heal Sandor "The Hound" Clegane from his wounds, and his volatile personality. The Hound is dead, he said, but what he meant was that Sandor Clegane was purged of his evil persona called "The Hound".
When I read this chapter, I get a sense of mystery and magic surrounding the Quiet Isle, but of course it is nothing as explicit as the miracles performed by R'hllor.
The stories told by the Elder Brother are fascinating. At one point, I was wondering if he was someone important, since he said he had been at the Trident when Rhaegar and Robert met, and that he "died" there, and was reborn on the Quiet Isle.
More on topic, it has been said that the Silent Sisters - an order of the Faith of the Seven - have the power to speak with the dead. But no such thing has been demonstrated, and moreover, the Silent Sisters have sworn vows of silence, so it would be a one-way communication.
In the books, the question on the existence of magic is a recurring theme. Most characters don't believe in magic, and the beginnings of the books don't lend much credibility to hedge wizards and the like. After a while though (end of first book/season) it becomes obvious that something is going on.
While the followers of R'hllor exhibit the flashiest and most powerful "magic", the other religions should not be counted out. In particular, the Old Gods and the House of Black and White seem to have followers with special powers. Furthermore, there are many other religions that may or may not exhibit these types of powers in times to come (Drowned God, Starry Wisdom, etc.)
With regards to the Seven, I think the strongest example yet actually isn't Quiet Isle, but is Davos' conviction that the seven kept him alive so that he could stop Melisandre's corruption of Stannis.
The "powers" that any of the religions have in the series can be best explained via Varys' anecdote about "the sellsword in the middle." The Faith of the Seven, in particular, gain their power from the people as the people perceive them as powerful. As far as any one religion being more powerful than an other, it's clear that there is a "force" (lack of a better word) within the metaphysics of the story that can be accessed by a character(s) consciousness.
Remember, GRRM hated the ending to LOST and proclaimed his disdain for it, so following that logic one can deduce that the "magic" won't be "other-worldly" and instead will revolve around a responsible agent, thus avoiding the writers cop-out of the Deus ex Machina.
One concrete example was Cats visit at Renlys camp and visiting the sept. There she was given guidance, but was too thickheaded to pick up the hints or act on them.
ie saw Arya as the warrior. But not in the stranger! (ie she will always be a someone ;) ) Saw Cersei as the mother Saw both Rickon and Bran needing their mother, and Robb requiring her presence too.
So yes, very subtle. No wonder one needs a clear conscience to be able to pick up the hints or act on them. And thus the punishments for sins are very much appropriate. Otherwise the seven are just painted idols to you.
No the clergy of the Seven have never displayed any magical abilities so far in the series.
The event about Hound has got to do more with the personal skills of the Elder brother rather than his magical powers. He got even someone as reserved and shy as Brienne to open up to him. He is a former warrior who found his peace at the monastery so naturally other lost warriors such as Sandor and Brienne can relate to him in certain manners, in order to give themselves hope of peace if nothing else.
As for Silent Sisters' powers to talk to the dead, while admittedly it has been stated in the books, it has never been stated as a fact. It has always been a rumor, a fantasy.
There have also been two instances of clergymen, who supposedly performed miracles. One was Septon Murmison, who if could have performed miracles by praying, might have done himself a lot of good by praying for the right temperament for managing the realm as Hand of the King. The second one was the eight years old boy who was made High Septon on King Baelor's insistance, as the King claimed that the boy could perform miracles. Yet, when the King had finally brought himself close to death in his final fast, the boy couldn't perform a miracle to save his life.
TLDR, there is no evidence that the clergy of the Faith have any magical powers. And given the lack of evidence, we can assume that they in fact have no powers at all.