Continuity mistake: Frasier (and later the others) come back to his apartment to avoid the rain - they shake off umbrellas, and we even hear the storm at the very start of the scene. However there's no rain on the balcony - can't be covered, as we see in "call me irresponsible" during the obligatory sex scene that when it's raining the balcony gets soaked.
Continuity mistake: Frasier tells his father Maureen wants to break up with him, and Marty heads for the kitchen, leaving his blue container of stomach medicine on the credenza. But as Marty passes by the dining room table, he suddenly has the container in his hand again.
Continuity mistake: In the conference room, Frasier says, "I just said that," and the shot shows his arms resting on the table with his hands up, gesturing. The next shot, from the side, shows him leaning back in his chair with his arms right off the table. The shot then goes back to the original position. There was no time for him to move naturally.
Continuity mistake: Near the beginning of the show, Frasier brings Roz a muffin in a bag which she places to the left of her phone (her left). During the scene, the bag shifts from being near the console to being back to left of the phone. At one point Frasier and Roz leave the booth, then come back in, and it has moved in the meantime.
Continuity mistake: Frasier and Martin are in the kitchen and Marty gets some chips out of the cupboard next to the fridge. As he leans down to pet Eddie we can see a bag of blue chips still in the cupboard - but when he straightens up to talk to Frasier, we see a red bag in the cupboard, not the blue one (the camera angle remains the same).
Continuity mistake: Frasier is sitting on the couch watching Wheel of Fortune and eating, with a plate on his stomach. As he talks to Niles he places the plate on the coffee table. Niles says "Are those profiteroles?" and Frasier gobbles them up. Then we see Frasier put the plate on the table again.
Plot hole: Niles phones Frasier to tell him Daphne is staying at his place because of the storm. Frasier's phone is in its usual place on the room divider/bookcase behind Marty's chair. After Frasier and Marty leave to rescue Daphne, Niles phones again and Eddie is standing up on a chair staring at the phone. Only problem is, the phone is over on Frasier's desk, where there normally is no phone. If he did have a phone there, it is only a few feet from the other one, which makes no sense.
Continuity mistake: During the first episode of Frasier there is a painting of some vases in Frasier's apartment which is at the end of the hall leading on to Daphne's room. In this episode the painting has a red and purple background, however in 'Here's looking at you' (S1, 2 mins 52 secs) the exact same painting is now red and blue. It remains this new colour for the entire 11 seasons.