Continuity: When Roz and Frasier are exchanging gifts, Roz takes her present out of its box and the box is still at least half covered in wrapping paper. In the next shot, no wrapping paper is visible around the box.
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Quotes
[Frasier tries to convince Niles that sleeping with Nanny G is the right thing to do.]
Niles: Did you say something? Your penis was talking so loud I couldn't hear!
Trivia
At one stage, Niles refers to Frasier's 'Big red clown shoes'. Kelsey Grammer provided the voice for Sideshow Bob in The Simpsons, a clown character who wore such shoes. In fact, in the first episode Kelsey Grammer's voice was used, those shoes were key to the plot. See more...
Frasier (1993) - 4 mistakes in "Miracle on Third or Fourth Street"
starring David Hyde Pierce, Jane Leeves, John Mahoney, Kelsey Grammer, Peri Gilpin (add more)
Updates: 
Series 10
Miracle on Third or Fourth Street (season 1, episode 12)
Continuity: When Roz and Frasier are exchanging gifts, when Frasier first puts the lid back on the gift box there is no tissue paper sticking out. In the next shot, a large sheet of red tissue paper is outside the box.
Continuity: Early in the show Frasier is wrapping a Xmas present for Roz. The shot cuts to Chopper Dave and then back to Frasier. A bow and gift tag have appeared on the gift, but there wasn't enough time for that to have happened naturally.
Continuity: When Bulldog is trying to persuade Frasier to cover his shift on Christmas Day, Bulldog's cup of eggnog jumps back and forth from sitting on the desk to in his hand.
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