Revealing mistake: The man from Mars is in an elevator with a woman when the power goes out. He tells her he cut the electricity and gas powered engines all over the world for a half an hour. They proceed to show clips of the power out in different parts of the world. When they get to Big Ben in London, you can see a motorboat cruising down the river in the background. If the boat were powered by wood or coal, there would be plume of smoke trailing it. If it were powered by oil (ie. diesel) electricity is required to pump the fuel into the ejectors.
Plot hole: Klaatu is shot and his body is lying in a prison cell. Gort leaves the flying saucer and walks all the way to the heavily guarded police station in downtown Washington DC. Gort is 8 feet tall and is a robot plus he uses his noisy death ray to create a hole in the police station to retrieve Klaatu's body and nobody noticed him? On top of that, he walks all the way back to the saucer unseen with a dead body.
Factual error: In the very beginning of the movie broadcasters around the world are commenting on the mysterious spaceship. To lend a bit of calm and normalcy one broadcaster notes that despite what is happening many people are enjoying the fine spring weather in the Capital, to this end picnics and the wonderful cherry blossoms are plainly depicted. Shortly afterward when Klatu leaves the hospital, he is seen examining the cleaner's tag on the suit he "borrowed". This is where he adopted the name Carpenter. Clearly printed at the top of this tag is the date 7/18 1951, not spring by at least a few months.
Continuity mistake: During the news broadcast, the TV announcer is evenly lit. You can see this in the side view of him, as well as on the TV monitor that's next to him. Then, just after he says "there is no reasonable cause for alarm," the microphone in front of him is suddenly casting a big shadow on him.