The clever idea to make a dart gun of a trumpet mute was not conceived by a musician. A mute cannot make an air-tight fit in the bell of the trumpet, as no air would pass and hence no sound; an air-tight fit would be necessary, of course, for the mute to act as a blow-gun. When Miss Fisher shows the mute to the inspector, you can see the three slender cork spacers on its neck that prevent an air-tight fit and allow the mute to function as a mute--but not as a blow-gun.
Collins mentions they received a telegram from the police department in Jamestown, VA. While it is now an historic site, no one has lived in the town of Jamestown since approximately 1700, when it was virtually abandoned for the nearby town of Williamsburg.
In the night club, the banjo player appears to be strumming, but his hand is inches off the strings and not moving in rhythm.
The biplane used in this episode (and correctly identified in the script) is a De Havilland Tiger Moth. The characters state that the airplane had been in a hangar "for years." The given year for this series is 1928; this is rather impossible as the Tiger Moth's first flight was on 26th October 1931.
The photos of the two lovers were taken in quick succession, which would have been impossible with cameras and flash equipment of the era. The subjects would have been able to move out of range before the second shot.
The plane used is a De Havelland DH-82A, manufactured in 1942. The series is placed in the 1920s.
They mention one entry wound but no exit wound. Why was there no open surgery autopsy? They would have found the dart.