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.
Continuity mistake: Just after Klaatu gets shot, Gort is seen melting the clear glass/plastic/cement material he was encapsulated in. When Patricia Neal begins to approach the space ship, the camera shows Gort still melting the material with it almost down to his ankles by this time. The next shot shows Ms. Neal looking at him, but when the camera goes back to Gort, the cement he was encased in is suddenly back up around his waist. How did it get back up when moments before it was down around his ankles?
Continuity mistake: After the power comes back on, a newspaper front page has Klaatu pictured with the president standing just outside the entrance to the saucer's dome, and Gort standing about five feet in front of Klaatu in the same place. But early in the film, Klaatu was on the ground with the president, and Gort never preceded Klaatu coming out of the saucer.