Continuity mistake: Daphne is on the phone with her mother and has a dust rag in her hand. She holds out the phone towards Frasier and when the shot goes back to her, the rag is all bunched up in her hand - it has been loose before. There was no time for her to change it.
Continuity mistake: Niles and Frasier are at Marty's old apartment and Niles sits down on the couch. Eddie is sitting next to him, staring. During this scene, the shots of Eddie show him moving from fairly close to Niles, to about 10 inches away, and then back again. He has no time to change position naturally.
Continuity mistake: When Frasier turns up to the office on his first day, having never met Roz before, he has the briefcase that Roz got for him as mentioned by Roz and Niles 'Fool me once, shame on you, fool me twice...'. How can this be if they've never met before?