Just before the officer candidates take the oath and graduate a recruit announces, "All officers present." The candidates' swords are up on their shoulders but in the next shot the swords are in their scabbards.
After Foley pulls Daniels out from the pool, his medal ribbons are missing their devices (little stars etc.). They are present before he jumps into the pool. It's unlikely they would have just fallen off.
When Gunny Seargent Foley has Zack Mayo doing push ups in the sand pit, it's noticeable that his forearm sleeves are already wet from getting into a prone position to chew the recruit out, before he actually does it.
The beginning of the movie is set in Seattle, Washington, but when Zack gets on his motorcycle to leave from his father's house, it is outside the Naval Base in Bremerton, Washington, on the other side of Puget Sound. Seattle does not have a Naval home port.
During the graduation ceremony the students are wearing the Service Dress White uniform and the reviewing officers are wearing Service Dress Blues. In reality they would be wearing all the same uniform one way or the other but never different as uniform season requirements are dictated by a senior officer (usually an admiral in control of the Naval region).
During the factory cafeteria scene where Paula is trying to reach Zack on the phone, a co-worker mentions to Lynette that it's 3:40 PM on a Friday and she's not bragging about their weekend plans. A moment later when Paula runs from the cafeteria to find Zack, the clock on the cafeteria wall reads 12:50PM. Also, many employees appear to be eating lunch.
The goof items below may give away important plot points.
During the graduation ceremony, when Gunnery Sergeant Foley reports to the officer and falls out the graduates, he does an about face and steps off with his right foot. This is incorrect, as you always start with the left foot when marching.
The class, upon receiving their commissions, approach Gunnery Sergeant Foley and present him with a silver dollar, in return for their first salute. When Ensign Seeger receives her salute, Sergeant Foley says "Congratulations Ensign Seeger, Sir". He should have said, "Congratulations Ensign Seeger, Ma'am."
When Zack and Paula go to the motel to find Sid they knock on the door with no answer. Zack goes to open the door with nothing in his hands. Cut to inside showing Zack and Paula going into the room through the door. Now Zack has a credit card and wallet in his hands indicating he probably jimmied the lock with the credit card to gain access.