Continuity mistake: Jason watches the broken statue with his face perfectly lit. A shot later he is looking downwards and his face is in the shadows. A shot later it swaps to the previous position with a fully lit face. (00:14:33)
Revealing mistake: After Jason's fight with Acastus, when Medea is telling Jason of the flower which can heal his wound, if you look past Medea you can see a mountain in the distance. On top of the mountain you can see a lighted building of some sort that appears to be a modern structure. (01:15:05)
Continuity mistake: When they arrive on the island of Talos, the bronze giant, Hercules and Hylus find the statues atop huge plinths. In close up, they are looking up at the statue of Talos, then in a wide shot they start to walk towards it across bare rocks, then in close up they are back where they were with plants behind them.
Continuity mistake: When Jason and the Argonauts pass through the clashing rocks, the necklace of Poseidon around Jason's neck changes from shot to shot, sometimes it's not there, sometimes it's tighter around his neck, other times it's a long necklace.
Visible crew/equipment: When the hydra comes out of the cave,look closely at the ground. The track the hydra is being pulled across is visible.
Continuity mistake: Talos changes size dramatically at several points in this film. For example, when he picks the Argo up, his head and feet are not much smaller than the whole ship, but when Jason is unscrewing his heel his entire foot doesn't seem much bigger than Jason alone.
Revealing mistake: Towards the end of the film Medea gets shot by an arrow. If you look closely you can see that no arrow is visible in the air, and that it would have to pass through Jason to actually hit her where it does, as he is in the way.
Continuity mistake: At the start of the final battle with the skeletons, Jason and his companions draw their swords and release the shoulder straps that are holding the scabbards, allowing them to fall. During the close-ups of Jason before the skeletons attack, his scabbard strap is back where it was.
Continuity mistake: When Talos is holding the Argo at least 100 feet in the air, we are treated to several shots of men falling out and you can see the boat is still sitting on the water.
Continuity mistake: During the fight with the skeletons, just before the first Argonaut is killed, he is about to climb a ledge and puts his sword down. Once he has been stabbed the shot changes, and you can see the sword is no longer on the ledge where he put it.
Continuity mistake: When Talos comes after the Argonauts, he reaches his hand through an archway in the rocks. After he withdraws his hand (in Harryhausen stop motion), notice one of the Argonauts wriggling and struggling against the rocks as if still trying to get away - the actor doesn't know what the end shot will be like and is still hamming long after the hand is gone.
Audio problem: At one point, I think it's when they arrive at the dead Hydra, Aeetes' Captain (with the brown "bobble-hat" and the moustache) says a line like "Some of you up there....and that way," and is pointing to direct the soldiers. It is very obvious that he is not saying this line and that it was dubbed on afterwards.
Revealing mistake: After Talos destroys the Argo, Jason swims to the statue of Hera. It is obvious he is in a pool because the water colour has changed, the water is lapping slightly (like a pool and not the ocean) and finally Hera seems to be propped up by something as she isn't moving with the tide.
Visible crew/equipment: The very first shot of a Harpy attacking the blind man is not up to the high standard of the rest of Harryhausen's work in this film - you can easily see two wires suspending the model as it "flies" down. These are better hidden in later shots.
Continuity mistake: When the giant bronze statue, Talos, attacks the Argo, the crew flees across the beach and up the ship's boarding ramp, and one sailor chops through a mooring line so the Argo can escape. But, in subsequent shots as the Argo departs, no chopped-off mooring line remains on the wide-open beach, and there's nothing to which they could tie a mooring line, anyway.
Continuity mistake: Jason loses a sandal when rescuing Pelias, but when he walks to Pelias' camp he still has two sandals on.
Continuity mistake: Just before Hercules throws the discus he folds his arms, but in other shots they are not folded.
Continuity mistake: When the archer shoots at Medea the sky is cloudy. When the arrow hits her it's sunny. This swaps back and forth.
Continuity mistake: After Jason stabs the hydra he leans on a rock placing either one or both hands, depending on the shot. This changes back and forth.
Continuity mistake: When Jason stabs the hydra, he retrieves his hand and walks away. When the shot changes, he's next to the hydra with his hand extended, and walks away again.