At the end, where Miss Flitworth dies — and Death takes her into the past to see that the man she loved didn't jilt her but was killed in an accident on the way to their wedding. Reunited, the ghosts go off into the afterlife together.
Slightly earlier, he brings her "a diamond to be her friend", misinterpreting "diamonds are a girl's best friend". Something about the scene tugs at my heartstrings in a wonderful way.
And it's implied he even earlier saves a little girl from death by giving her some of the sand from his own life-timer, thus bringing his own death (which terrifies him) closer.
What really sells it is the little internal conflict between Death and Bill Door. Death knows that children die, not in a cruel or uncaring way, but just as a fact of existence. To Bill Door "This was all so much horse elbows."
There's a subtler one earlier where the farmers remark on how Death cuts corn very quickly despite cutting each stalk individually. This evokes how he empathizes with humanity and sees them as individuals, unlike the New Death later in the book who treats them like cattle.
What can the harvest hope for, if not for the care of the Reaper Man?
For the sake of prisoners and the flight of birds.
And he stops and has a chat with Windle Poons. This a kinder Death.
As Death sharpens his new scythe.
Miss Flitworth: Silk. Finest white silk. Never been worn.
Death: ... Thank you.
The 'Going Away' party the members of UU had for Windle, especially after he explicitly said he was upset that no one seemed to care that he was about to die.
A minor one when the exiled Death meets the Death of Rats for the first time.
Death:I remember...when you were a part of me.
The fact that Death beats the New Death not with his new super-sharp scythe as he had planned, but with the ordinary worn scythe he had used for the harvest.
Death creating golden fields of corn around his house at the end of the book, the only thing in his domain with any colour besides black or white.
Or purple. There was purple there too in "Mort"
That the last thing the Biography of Windle Poons will record, in Death's library, will be Ludmilla and Lupine running through the hills together next full moon.