Revealing mistake: At the river to cross over to Mt Weather, the vine used to swing across the river is just a standard rope, camouflaged. You can see it best when Finn recovers it after Jasper has jumped.
Murphy's Law - S1-E4
Continuity mistake: When Clarke and Finn are undressing in the bunker, right after Clarke takes her shirt off in a closer shot they both have bare shoulders. In the following shot Finn has his shirt on once again before pulling it over his head. (00:39:30)
Contents Under Pressure - S1-E7
Continuity mistake: When Clarke extracts the knife from Finn's chest, suddenly they all get thrown to the ground. Once she stands up again, in the first shot she's holding the knife with the blade pointing behind her, as it was when she extracted it, but in the following shot the blade is pointing forward instead. (00:14:20)
Continuity mistake: When Clarke gets in Finn's tent with Raven to check on his wound you can see the cut running vertically on his chest, but during the whole previous episode the wound was horizontal. (00:06:35)
Revealing mistake: When the guard slices Shumway's wrist, you can see blood coming from behind the glass shard instead of from Shumway's skin.
Plot hole: The grounder's leader Anya states that the missiles (flares) burned a village to the ground. If this were true, the flares couldn't have flown very far, otherwise that village would have been too far away for the grounders to know about. In a previous episode, when you saw the flares from space, it was obvious the flares traveled hundreds if not thousands of miles away. (00:31:30)
Factual error: Bellamy tries to fire his assault rifle twice, but because of a dud round, it fails both times. There is an audible click each time he tries. The first attempt there would be a click, because the firing pin was charged when the last round was ejected. However, without that charge either automatically when a round is ejected or manually pulling the charging handle, the second attempt would have been silent since the firing pin was not engaged. (00:35:45)
We Are Grounders: Part 2 - S1-E13
Factual error: After the Arc goes to earth, the chancellor is sitting in a floating ring in space that no longer has any centrifugal motion. He should be floating, but he is seated and gravity is apparent.