Factual error: In surgery Trapper John is singing Frank Sinatra's version of "I got you under my skin". Although it was written in 1936, Sinatra did not release it until 1956, after the Korean War ended. The 1936 version sung by Al Bowlly sounds nothing like the version Trapper John was singing, which was mimicking Sinatra's version.
Suggested correction: While it hadn't been released on vinyl until 1956, Sinatra had sung the song as early as 1946 on his radio show and during live shows.
The version Frank Sinatra sang on his radio show was similar to the original version used in the movie "Born to Dance" (a movie he references before singing his two songs). He didn't change it to the big band version that Trapper imitates until 1956.
Continuity mistake: When Frank walks into the Swamp and puts down the typewriter, in shots facing Frank the manila envelope's open flap faces up, but in shots facing Trapper and Hawkeye while they're pretending to argue the envelope is flipped over with the flap facing down, then when Frank walks back to the envelope the flap faces up.
Continuity mistake: While Hawkeye and Trapper are playing shotglass checkers Frank lathers shaving cream onto his face, but in his closeups the thick foam is dried out.