In the books we learn that Joffery Baratheon, most likely, sent the catspaw to kill Bran Stark in an attempt to impress his "father", Robert Baratheon. This is first worked out by Tyrion Lannister:
He remembered a cold morning when he'd climbed down the steep exterior steps from Winterfell's library to find Prince Joffrey jesting with the Hound about killing wolves. Send a dog to kill a wolf, he said. Even Joffrey was not so foolish as to command Sandor Clegane to slay a son of Eddard Stark, however; the Hound would have gone to Cersei. Instead the boy found his catspaw among the unsavory lot of freeriders, merchants, and camp followers who'd attached themselves to the king's party as they made their way north. Some poxy lackwit willing to risk his life for a prince's favor and a little coin. Tyrion wondered whose idea it had been to wait until Robert left Winterfell before opening Bran's throat. Joff's, most like. No doubt he thought it was the height of cunning.
...
The blade Joff chose was nice and plain. No goldwork, no jewels in the hilt, no silver inlay on the blade. King Robert never wore it, had likely forgotten he owned it. Yet the Valyrian steel was deadly sharp . . . sharp enough to slice through skin, flesh, and muscle in one quick stroke. I am no stranger to Valyrian steel. But he had been, hadn't he? Else he would never have been so foolish as to pick Littlefinger's knife.
The why of it still eluded him. Simple cruelty, perhaps? His nephew had that in abundance. It was all Tyrion could do not to retch up all the wine he'd drunk, piss in his breeches, or both. He squirmed uncomfortably. He ought to have held his tongue at breakfast. The boy knows I know now. My big mouth will be the death of me, I swear it.
A Storm of Swords, Tyrion VIII
In the books, as quoted above, we also learn that Joffery hadn't really known about Valyrian steel at the point he sent the catspaw. The only quote I can find in the show that resembles this, and it is a bit far-fetched, is the following:
Careful, Your Grace. Nothing cuts like Valyrian steel.
So they say.
Game of Thrones, S04 E02, "The Lion and the Rose"
With Joffery's cocky attitude it could imply that he already knows about Valyrian steel at this point although not really having come into contact with it before. Except Ice when it was taken from Ned Stark but I interpret that comment to be more aimed towards the dagger in a backward kind of way to reference the quote from Tyrion in the books.
Later on Jaime and Cersei Lannister also appear to come to the same conclusion:
Robert? Jaime had guarded the king long enough to know that Robert Baratheon said things in his cups that he would have denied angrily the next day. "Were you alone when Robert said this?"
"You don't think he said it to Ned Stark, I hope? Of course we were alone. Us and the children." Cersei removed her hairnet and draped it over a bedpost, then shook out her golden curls. "Perhaps Myrcella sent this man with the dagger, do you think so?"
It was meant as mockery, but she'd cut right to the heart of it, Jaime saw at once. "Not Myrcella. Joffrey."
Cersei frowned. "Joffrey had no love for Robb Stark, but the younger boy was nothing to him. He was only a child himself."
"A child hungry for a pat on the head from that sot you let him believe was his father." He had an uncomfortable thought. "Tyrion almost died because of this bloody dagger. If he knew the whole thing was Joffrey's work, that might be why . . ."
A Storm of Swords, Jaime IX
However, the wikia for the show seems to suggest that Petyr Baelish was the one who sent the assassin:
During the trial, it is deduced that the dagger did in fact belong to Baelish all along, suggesting that he was the one who sent the catspaw assassin to kill Bran in the first place.
The quote this appears to go off of is the following:
You told our mother this knife belonged to Tyrion Lannister. But that was another one of your lies. It was yours.
Game of Thrones, S07 E07, "The Dragon and the Wolf"
This appears to be a bit of a stretch and there's no actual evidence that Petyr owned the dagger beforehand, the wikia doesn't even state him in the list of owners before the catspaw. It appears as if Sansa is saying it as too add to the "evidence" for why Petyr is going to be given the sentence he is given. Even if the dagger was Petyr's it doesn't even mean that he sent the catspaw too.
So, do we ever learn who sent the catspaw?
Note: I've tagged with the books as my question uses evidence from the books but I am interested in show only answers.
