Hawkeye refers to his mother and sister in the letter to his father. Later in the series, it was established that Hawkeye's mother died when he was young and he was an only child.
During the song "My Blue Heaven" Radar begins playing his drum kit with brushes. In the next shot he now is playing with sticks. When the song finishes he is playing with the brushes again.
In addition to the previously cited discontinuity of Radar changing from brushes to sticks and back to brushes during the "My Blue Heaven" performance, the piano players change as well: a soldier with black, wavy hair (under cap) while Radar plays with brushes; to a different soldier with slick-backed hair, when Radar is playing with sticks; and back to the first piano player when Radar is playing with brushes again -- further confirming that this performance is composed of two separate takes.
During the shows epilogue while Hawkeye is saying goodbye to Captain Casey, Casey is wearing tortoise shell eyeglasses for the head-on shots but metal framed specs for the over-the-shoulder shots of Hawkeye.
When walking into the mess tent "naked," not only can you see Hawkeye wearing boxers, but they change. When he first goes in, they are army green. When Trapper comes in, they are white.
In the first OR scene, when Hawkeye is checking Capt. Casey's surgical work, the camera is shaken to simulate an artillery concussion a split-second before the sound of an artillery explosion is heard.
When Goldman drops the food tray he does not act startled, shocked or surprised but instead intentionally and purposely releases the tray from both hands at the same time.
During surgery, while the doctors are all bickering, Col. Blake yells at them to knock it off. At this point he is looking at the doctors, but just as the scene ends, he looks right at the camera.