The map of Germany and the Low Countries that Stokes and his men look at shows a map with modern day coastlines, including the Dutch province of Flevoland. However, the 1,000 square kilometer insular province only came into being when it was reclaimed from the sea in 1968, well after the Second World War. Also, the "Afsluitdijk" (a 32km long dam sealing off the Zuiderzee) is not shown on the map, despite having been completed in 1933.
When the German trooper says "John Wayne" as a household name, it is unlikely the average GI would make a connection. Wayne made B Pictures for years after Stagecoach (1939) and would not achieve international fame until the end of the 40s.
Frank Stokes comments on Hitler that he was a "failed Vienna art student" during the briefing to his team. However, Hitler was not an "art student" at all in the beginning. His heart was in the architecture but he couldn't get into Academies School of Architecture because he did not finish high school. Instead he applied to the Academy of Fine Arts Vienna (which did not require high school diploma) but was rejected there as well. So he made living as a wannabe artist selling hand-painted postcards.
When Lt.Stokes turns around from viewing the slide of the map of France, he has a pencil in his hand and the clip board is on the table. In the next shot from over his shoulder, the clip board is in his hand and he then places it back on the table.
During live fire training, the location of the World War I memorial is attributed to Saint Louis, Missouri. The sculptor upon whom John Goodman's character is based designed sculptures for St. Louis' World War I memorial, which is called the Soldiers Memorial and should not be confused with the National World War I Memorial located in Kansas City, Missouri.
One of the Monuments Men is taken for a ride on a biplane that has been hidden in a French barn. The aircraft registration number starts with a G, indicating that it was not registered in France, but in Great Britain.
When US Army Lieutenant James Granger visits the apartment of Claire Simon, he is not wearing his uniform but is wearing a casual, short-sleeve shirt instead. During World War II, it was a strict rule that members of the military had to wear their uniforms at all times, on and off base, even while home on leave. Liberated Paris in 1944 was still considered a war zone, making his disregard for uniform especially egregious.
Prior to the scene where Generals Patton, Bradley and Eisenhower are seen in a "photo-op" involving recovered treasure, a row of parked jeeps is shown with stars on red bumper plates belonging to the generals. One jeep (presumably Eisenhower's) has five stars in a horizontal row, whereas the actual rank insignia and bumper plate would have the five stars arranged in a ring with an empty pentagon-shape in the center. Also, that jeep and the one beside it with four stars (presumably Patton's) both have bumper numbers that begin with "3A" indicating both vehicles are assigned to Patton's Third Army (as are most of the vehicles used by the Monuments Men themselves); as Supreme Commander of the European Theater of Operations, it's highly unlikely that Eisenhower would have had to borrow one of Patton's jeeps.
When the German asks the French assistant for a champagne glass, the woman gets what would be a champagne glass if they were in America; champagne glasses in France are tall and thin, not squat and wide-brimmed.
The first mention of the Madonna is in connection with the Onze-Lieve-Vrouwekerk ("Church of Our Lady"). The church shown in the film, though, is a completely different one. Also, the Madonna is displayed in an altar piece, not standing on its own.
The "Nero Decree" issued by Adolf Hitler on March 19, 1945, never mentioned "archives and art". In the movie, however, when Stokes reads the decree aloud, he lists "archives and art" among the things set to be destroyed.
Matt Damon's character stepped on a landmine but in reality Lt. James Rorimer didn't. Most German mines were never landmined although entrances and tunnels were sealed by blowing them using normal explosives before the arrival of the Allies.
George Clooney is shown as the only person finding the Madonna of Bruges statue and then the only person pushing the mine cart with the statue in it before Matt Damon arrives to help him. The Madonna of Bruges statue is marble and is 200cm (76") high and would be impossible to lift and move by one person. Wikipedia has a photo of the statue in the Altaussee salt mine being moved with at least 5 people in the photo.
The goof items below may give away important plot points.
When old Stokes comes back to Bruges to see the Madonna in 1977, he can come close to the statue to view it. The statue has been behind bulletproof glass since 1972 and the public can not come closer than 15 feet.
When the Frenchman Jean-Claude is shot, the Army unit present would have tended to his wound(s). Additionally, the Monuments Men would have had a first aid kit in their truck to use until they could get to an aid station.
During Stokes' proposal, he shows a colour slide of the Ghent Altarpiece, in which it appears complete. In 1934, the bottom left panel ('The Just Judges') was stolen and has never been recovered. It was replaced with a copy in 1945. It is highly unlikely that Stokes would have a colour slide more than 10 years old of the Altarpiece. At the end of the movie, the black and white slide he shows of the returned Altarpiece is a contemporary photo correctly showing the missing panel.