When Niles dumps the ashes from his shoe in the church, the edge of the carpet runs under where he is seated, and the ashes fall half on the wood floor, and half on the carpet. When the janitor cleans up the ashes after the service, the carpet stops at the end of the pew, and the ashes are completely on the bare wood floor.
After Frasier opens and spills the urn, it's still full but on the next shot that shows Martiin's reaction, it's completely empty.
As Niles is transcribing Martin's song onto paper, a piece of printed sheet music is seen on the piano instead of a piece of empty staff paper.
At the park, when Frasier and Martin are talking in the car, a piece of audio/visual equipment is fully visible in the center of the car. It's neither a typical overhead light nor a rearview mirror so it's obviously a function of the show's set.
In his apartment, the highly fastidious Frasier once again chides Martin for Eddie's lounging on the sofa, but says nothing about Daphne cutting Martin's hair right there at the dining table! Daphne even brushes hair off Martin and nothing is said or done about cleaning up the hair clippings.