After the man left the house driving a 1930s car, the woman told the taxicab driver "Follow That Volkswagen" in closed captions. This car was not in USA until 1949.
The tail of the plane photographing the Queen Anne is visible in the lower right hand side of the picture.
When William Powell is practicing fly fishing in his hotel room, the lamp is knocked over by a visible wire.
When Gladys storms into the newspaper office to confront Haggerty about not turning up for their wedding, He tries to send her away saying 'he'll join her as soon as he's settled the suit' At that point in the movie they do no know yet that there is even going to be a suit to settle.