Goofs

When LeBeau is leaving the barracks without a gun, he ducks as the guard's lights shine at him. When he comes back up, he has a gun.

Update: When he leaves the barracks, he is wearing the coat he wore earlier, which appears to be a size or two too large. Since LeBeau waited until all the other prisoners were asleep before leaving, it is entirely logical that he would hide the gun inside his coat, and not pull it out until after he left the barracks. There is plenty of time between the time he ducks down until the time he is shown getting up for him to pull the gun out of his coat. Later on, in Wilhelmina's apartment, he puts the gun in the front left pocket of the coat, and the gun is never seen again.

Twice, Schultz drives off without Klink in the car. In following scenes, Klink is in the car; Schultz had to go back to get him both times, but this is not shown.

When Schultz drives off and leaves Klink behind the first time (when leaving Stalag 13 to go get LeBeau), there is a sound of screeching tires as Schultz hears Klink and stops to go back to him. Since there is no paved road in the camp (the ground is just packed earth), there would not be any screeching tires. There might be some sound as the tires rub on the dirt, but no screeching tires.

LeBeau has been made aware that the Gestapo have captured a major underground connection, which results in his attempted 'rescue' of his contact, Wilhelmina. Both of them should have known that, once the Gestapo had neutralized a high-profile player, there would be an all-out effort to track down other members. Yet, Lebeau watches as Wilhelmina repeatedly attempts to make radio contact, thus drawing the Gestapo right to her building.

When Hogan and LeBeau are talking together in the barracks, there is a sign, in German, posted on the wall behind them reading, "Dieses Haus ist für Schützen", which translates to "This house is for shooters". This makes no sense, since Hogan and his men, as prisoners of war, would not have access to guns or other weapons. Secondly, signs and notices inside the prisoners' barracks would logically be in English so the men could read and understand them.

There are cases where there is a production light reflection in Klink's monocle.

Wilhelmina would not have used her real name on the air as she makes radio contact with them. She would have used a code name to add some level of security in the event the enemy was listening in on the transmissions.

As leader of the unit, it makes no sense that Hogan would not have met (or at least seen a picture of) Wilhelmina in order to ensure the security of his people. Standard procedure would be for him to be the first one to meet any new contact to evaluate the security arrangements and establish the contact protocols and passwords (which seem to be non-existent) - in case another member of the team needs to meet the contact.