The 43rd, a Moving Story - S1-E23
Other mistake: When Hogan uses the water barrel periscope to spot Manor Kuhn, the first shot is from an impossible angle.
The Late Inspector General - S1-E4
Other mistake: While the General is in Klink's office making the recommendation for Klink to be commandant of all German POW camps, he mistakenly calls Stalag 13, Camp 13 three different times. Camp 13 was referenced in the black and white pilot episode.
Other mistake: Hogan is disarming a booby trap briefcase. The bomb expert tells him over the radio he has to decide to turn the handle clockwise or counterclockwise. The expert doesn't know which way to turn it. Hogan turns the handle counterclockwise and disarms the booby trap, and the expert says counterclockwise. There's no way the expert would have known which way Hogan turned the handle. (00:21:50 - 00:22:26)
Factual error: While introducing himself to the downed Russian pilot, Hogan claims he is with the "US Air Force." He was actually with the United States Army Air Corps (USAAC) as the US Air Force wasn't formed until 1947, 2 years after the war ended.
Oil for the Lamps of Hogan - S1-E14
Plot hole: The Heroes take an empty oil drum to the depot, knowing full well that Schulz will protest, allowing them to steal a full one. However, all but their barrel have snow on it. Since it's really salt, it all falls off as they load it on the cart. With real snow, that wouldn't happen. It would be quite obvious that the barrel they are taking out is not the one they brought in.
Plot hole: Five minutes into the show, Klink and Hogan are in Klink's office when a gestapo major comes in. When Hogan leaves, Klink mentions Hogan's name and the major realises with surprise that that is Hogan and then says he knows all about Hogan and his description. That alone should have led the major to know that this was Hogan. Hogan also has his name on a tag on his jacket.
German Bridge Is Falling Down - S1-E7
Continuity mistake: During his experiments to create explosives in the tunnels, Carter has a pretty extensive set of glassware on his workbench. Necessarily he would have to replace (most of) that after each explosion - not to mention various light bulbs, furniture and other non-blast-proof stuff by the way. A lot of that glassware is specialty equipment, it would not be easy to come by even one set of in peacetime for a free civilian. In wartime, for an allied prisoner (even with the heroes' connections) it should be nigh impossible, and totally impossible to have an inexhaustible supply of the stuff. So we can either assume a giant plot hole, or treat it (as I did) as a big continuity mistake.
Suggested correction: Considering they regularly get equipment of all kind air dropped from "London", a few laboratory instruments and containers would hardly be challenge.
Firstly, considering the number of explosions, it's not "a few" but more like "QUITE a few." Secondly, if they had had the option to receive airdrops at the time, they would just as have had them airdrop the explosives instead of the glassware for carter to blow up, wouldn't they? Or are you suggesting they would have more spare laboratory equipment in store than a wholesale laboratory outfitter, "just in case"?
Happy Birthday, Dear Hogan - S4-E26
Continuity mistake: LeBeau meets a female underground contact in a movie theater. They have to kiss to avoid detection by the Gestapo. When LeBeau leaves the theater there are no lip stick kisses on his right cheek, but when he returns to camp he has some. (00:18:00 - 00:19:00)
Suggested correction: Simple, they kissed again before LeBeau arrived back at camp. Nothing to see here.
Other mistake: In the movie, as the jeep comes up to Byron Buckles' plane, the jeep rolls through previous tracks from previous takes.