Near the end, when Maria yells "Don't you touch him!", two different voices can be heard at the same time in the first half of the phrase (in fact, this is singer Marni Nixon overdubbing for Natalie Wood).
When Doc says, "What are you doing there?" after stopping the Jets from taunting Anita in the drugstore, his lips actually mouth, "What are you doing now?" which was the line in the original stage script.
The Jets sing "The Jets are gonna have their day tonight. The Jets are gonna have their way tonight". One of the voices sings "way" instead of "day" in the first line, so the result sounds like "dway".
In the opening number, which was shot on location in NYC, the playground is easily 300 feet x 300 feet in size. Later on, when it is obvious they were on the studio back lot, the playground has shrunk to around 30 feet by 30 feet.
Chino visits Maria on the rooftop, saying that there was no-one home. Maria tells that her parents are at work. Only a few moments afterwards we hear voices: "¡Mamá, Mamá! ¡Bernardo está muerto!" "¿Bernardo? ¿Muerto?", the latter apparently said by the mother.
This post is actually incorrect. Maria says to Chino that her parents are at the store, not work. In addition, the building is populated by several families, not just Maria and Bernardo's. Had it been their mother talking to the boy at the bottom of the stairwell, there would have undoubtedly be more of an emotional response from the woman as Bernardo was her child as well.
After the war council, Schrank says that he rebuked the Jets the same day. That means that all events before the war council were on that day too. The dance started at 10 PM, and that leaves very little time for America, the balcony scene and the war council.
At the beginning of the movie, behind the first close-up of Riff on the playground, snapping his fingers, you can see a young boy riding his bike behind Riff, the boy wasn't there in the previous shot (from above).
When we first see The Jets in the playground as they are snapping their fingers to the music, it is very bright and their shadows face towards the screen, but when they cut to the next shot as they start walking and the camera moves behind the fence, the sunlight is not the same as the scene is not as bright and even the shadows positions are different, pointing away from the screen.
Towards the end of the America number, there is a shot of a side view of the dancers and there is a metal structure to the right of the screen that has 3 people standing in front of it, one of them a lady with a blue skirt, but when the shot changes to the front and the structure is now in the back, those 3 people standing there have vanished from the shot, instead seeming to appears now towards the left side of the screen off-camera.
The goof items below may give away important plot points.
During the rumble (the big fight between the Jets and the Sharks), Bernardo gets cut in the back by Riff. During the fight, there is no cut on the back of Bernardo. Whilst lying on the ground after he's killed, there is a big cut across his back.