Plot hole: In several occasions throughout the story, e.g. S4E18, Hogan attends parties at Stalag 13 wearing an immaculate US dress uniform. Where did he get that? Dress uniforms are not part of the usual kit a bomber crew brings along with them on their missions.
Plot hole: Hogan has his men play romantic music over the radio. Earlier they were afraid of the radio in the observation post being detected. Elsewhere in the series, they try to avoid sending too long to avoid being homed in on - which is correct. Now, if they were to play a whole record at once, wouldn't the homing devices pick up on the transmitter at the camp?
Plot hole: The prisoners fake a fire with some smoke bombs to have an excuse to rush into the print shop and smash up the print plates. Afterwards they even receive thanks and a reward for their valiant efforts. In reality, the guards would notice immediately that nothing in that room is even singed, but everything important has been smashed by fire axes.
Plot hole: The boys take out the radio control unit from the mini tank. Le Beau is needed to get in and drive it to make it appear okay. After going between the buildings, they get him out, and start it on its way. Trouble is, no remote control and no driver, yet it turns and operates smoothly.
Plot hole: This episode revolves around training the reluctant, untalented Schultz to impersonate Klink to get the captured heroes back from the Gestapo. This is actually a pretty common theme, somebody, usually one of the heroes, impersonating an officer to free a prisoner. The plot gives no reason why this time, they would have to use the cowardly, untalented Schultz instead of doing it themselves.
Plot hole: When the driver of the car addresses Roberts (the real Roberts that is, not the double), Roberts reprimands him, telling him not to call him Baumann. It is safe to assume the guard was supposed to be speaking German, since he addressed Roberts by his German identity. That poses some interesting questions. Did Roberts speak English? In those days, most Germans did not even have a basic knowledge of the English language. Did he speak German? That would imply he has a good enough command of the German language to pass as a German. Roberts however isn't a spy like Hogan and his crew, but a pilot in the RAF.
Plot hole: We see Hogan's men empty out the gunpowder from a lot of cartridges inside the armoury. What did they do with their empties? It's not like a mound of empty cartridge cases on the ground wouldn't draw some suspicion is it? Yes, I am aware some options come to mind, e.g. putting the bullets back in and stuffing them back into the belts, but none of that is actually shown or talked about - LeBeau even chucks one cartridge over his shoulder, implying they are not very concerned about hiding their tracks.
Plot hole: Hogan has the men fly kites to get wind data and build a hot air balloon for a man to escape in. In reality, kites and hot air balloons are pretty much mutually exclusive - if there's enough wind to fly kites, one can't launch a hot air balloon, and vice versa.
Plot hole: In his German uniform, Col. Hogan wears a number of decorations, among them the Iron Cross first and second class. When did he earn those if he has just been drafted for a suicide mission? After all he was not posing as a Wehrmacht soldier for a change but wore the uniform "officially". (00:17:00)
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.
Plot hole: Hogan's men have that clever device in Klink's quarters, where they turn the oven from below to reveal the trap door. That however unplugs the oven from the chimney, you can even see the smoke pouring out of the oven in some shots. Often enough, they have the oven unplugged for half a minute or more. If an oven smoked into a room that long, the whole room would fill with smoke so badly it could actually lead to smoke poisoning. If nothing else, Klink would have the oven checked. Any device that can turn a heavy cast-iron stove flawlessly like that would by definition have to be quite massive and not disassemble easily. If Klink had the oven checked, that would certainly lead to the discovery of the turning mechanism.
Plot hole: In the cinema, Hogan and his female underground contact avoid attention from a passing plain clothes policeman by kissing. This is wrong for three reasons. Firstly, 3rd Reich Germany wasn't so liberal that open kissing in the cinema would have passed as normal, it would probably rather have been viewed as libertine and offensive. Secondly, the scene takes place with a speech of Hitler as the background, which can hardly be viewed as romantic, and thirdly, this is even more true for the Gestapo, who was tasked with ratting out political dissent. To a Gestapo officer, ignoring the Führer's speech and kissing instead would make them stand out doubly.
Plot hole: Hogan's men steal the motor from the silent plane. When Klink and Burkhalter inspect the plane, Burkhalter doesn't notice the engine missing. Later Klink doesn't notice the fact that it has been put back. This is totally impossible for two reasons: Firstly, if a propeller airplane misses its motor, there is no place to mount the propeller. Secondly, when we get to see the whole plane, it becomes obvious that it has an open engine cowling which leaves the motor exposed to be cooled by the airstream. The propeller issue aside, if there is enough light to see the hand in front of your eyes, it is impossible to miss either the presence or the absence of an engine inside that kind of cowling. Add to this the fact that both Klink and Burkhalter are Luftwaffe (Air Force) officers and therefore by trade have some experience with airplanes, the whole thing becomes even more ridiculous.