During the staff meeting, while Margaret is reading her divorce decree, she has a 1st Lieutenant bar on the left side of her collar, where she should be wearing a Nurse Corps insignia (caduceus with large black "N").
Nurse Bigelow is from Beaver Falls, Pennsylvania, which she says is close to Harrisburg. In fact, the two places are more than 200 miles apart.
When Hawkeye goes back to the mess table to get Margaret's attention, Loretta Swit's fingernails can be seen to be over an inch long. No nurse, especially a head nurse, would have fingernails that long.
Lt. Nugent casually accepts a date with Radar, a corporal. Fraternization (an officer dating an enlisted man) is a court-martial offense in the Army.
When Hawkeye is leaving the shower, you can see the top of the shorts he's wearing.