I couldn't resist and loked up one of my my favorite movies "Saving Private Ryan". They list 81(!) inconcistencies for that movie. However, that website is not entirely correct, some examples:
Factual error: A Hitler Youth dagger is found in the trench right after the first bunker is taken on the beach. The men in these bunkers were mostly old German soldiers (in their 40's), veterans of the Russian campaign and Ukrainian and White Russian conscripts. It's highly unlikely that a member of the Hitler Youth or an SS Division composed of them would have been manning the bunkers on that stretch of Normandy, which had not been expected to be an invasion target. These divisions were kept further inland, with the rest of the SS Panzer divisions
It is possible that the dagger belonged to a former Hitler Youth, many HY members who were old enough (18+) went on to become officers. So, no fact.
Factual error: The most obvious mistake to me is the German hair cuts. Most have completely shaved heads. War time photos show long (often very long) hair on top of their heads with only shaved portions around the sides. The movie hair cuts are more indicative of Russian or Japanese soldiers.
Wrong. Many soldiers shaved their heads for hygenic reasons. The lice were a real problem in the trenches. No fact here either, sorry.
Continuity: The interpreter says "Germans couldn't tell my accent, the might only think I have a small Bavarian accent," or something like that. When he then speaks German he has such a clear, bad American accent that even the dumbest German could tell that he is a foreigner
Wrong. Upham says "...and my German has a touch of Bavarian", which does not imply that he knows it prefectly, but indicates how he pronounces certain words.
Continuity: On the shore it shows a full can of ammunition floating in and out on the tide. A full can would be as heavy as a rock and would not move
Ummmm... How do we now that cans where full?
Factual error: In the scene towards the end where Corporal Upham is translating the French song 'Tu es partout', he translates correctly but his translation does match the song. He says everything prematurely, before it's actually said in the song.
Could it be that Upham knew the song and was translating it for the other soldiers? Sounds plausible to me. No fact.
Continuity: When the old Ryan is walking along the rows of graves, you see that his jacket is blowing open against his arm. It then cuts to a close-up, and the jacket is flat against his chest.
Couldn't the gust of wind cause that? Those guys are really nitpicking now.
Continuity: When the door of the landing craft opens in the beginning of the film, a soldier is immediately shot in the head. As he goes down his eyes close, open, close ... To notice it you have to look frame per frame. When he has his eyes open you see him check where he has to fall.
Frame by frame? Why????? This is just plain stupid. He's an ACTOR! Wouldn't you want to know where you're falling before taking a leap?