As Gary and Martha sit in their booth following their duet at the Cocoanut Grove, the champagne bottle is unopened and pointing away from the table and their glasses are empty. As the camera cuts to them sitting down, the bottle is opened, pointing towards the table, and their glasses are filled.
When Martha is singing over the PA system in The Green Room she sings "Canadian Capers' for approximately 7 minutes. 78 RPM records, the type playing, had a maximum play time of 3 minutes 15 seconds.
During a montage showing Martha's rise to stardom, there's a shot of stacks of fan letters, all bearing what turns out to be Martha's home address. Even in the far more innocent times in which the film was set, stars didn't publicly reveal their home addresses and virtually all fan mail would have been addressed to her movie studio or record label or simply "Hollywood."
When Doris Day moves in with Eve Arden, she's in bed making a phone call to here son in New York. It would be WAY past the boy's bedtime, since New York is three hours ahead of Los Angeles. It would be the middle of the night for the kid.
(at about 33 mins) Doug is coming out of the bedroom telling Vi: 'You can't say 'no' at a time like this,' but his lips and mouth movements are clearly saying something different.
When Martha is tucking Freddie into bed, a microphone shadow is visible on the back wall of the bedroom.