Sansa's Letter to Robb
I was just rewatching parts of the earlier seasons, so this has started to nag at me. It may be a plot hole, given that it's what drives Littlefinger's machinations in Season 7. Or it may just be an inconsistency.
Robb takes Sansa's letter with him when he goes south. We know this, because he later shows it to Catelyn when the two of them meet up. The only way it would have gotten back to Winterfell would have been if Ser Rodrick took it back with him when he returned north. Which is plausible, I suppose, but it seems more likely that Robb or Catelyn disposed of the letter at some point. Because it seems rather odd that Robb would give Ser Rodrick a private letter addressed solely to Robb to take north just so it would end up in the Winterfell library.
Also, the Ironborn set fire to Winterfell when they abandon it. And while the castle is stone, so not all of the rooms would have been torched, one thing they do beforehand is slaughter all the ravens. So if they go to all that trouble to kill the ravens, the library might be one of the rooms they would make sure to destroy, to remove all written evidence of their actions. So even if Ser Rodrick had brought the letter north, it may have ended up being destroyed anyway.
Littlefinger would have known about the fire, so he would have looked into whether the library had been destroyed. But it still doesn't explain why the letter would have been at Winterfell in the first place.