These are what we call the 'YMMV items.' Things that some people find in this work. We call them 'your mileage might vary' because not everyone sees these things in the same way. This starts discussions in the trope lists, a thing we don't want. Please use the discussion page if you'd like to discuss any of these items.
Complete Monster: Dr. Destiny, aka John Dee, transforms from a standard, somewhat campy supervillain into a nightmare inducing lunatic. When first seen Destiny comes across as a harmless, even somewhat pitiable, man. This impression goes out the window when he, for no real reason, casually murders the woman who had given him a ride, made pleasant small talk with him, and given him a coat. Once the good doctor regains the Materioptikon, a ruby used by Morpheus to create reality from the fabric of dreams, Destiny shows off just how bad he truly is. Sitting in a diner, over a twenty-four period Destiny begins steadily driving the establishment's patrons and employees mad. He makes them have sex with each other, mutilate themselves, and torture and kill each other. At one point he gives them their minds back and when they demand to know why he's doing this to them, he responds, "Because I can." By the time he's finished all of the diner's inhabitants are dead. All while this was going on, Dr. Destiny was using the power of the ruby to drive everyone on the planet mad. One disturbing instance had him making a kids' show host tell his viewers to slice open their wrists. When Morpheus asks Destiny what his goal is now, Destiny replies that at first he wanted to rule but now he wants to destroy the world and dance in the wreckage.
Ensemble Darkhorse: Lots and lots, though given the size of the cast that's not surprising. Lucifer, Death, Hob Gadling, Matthew, Mervyn, and Fiddler's Green all have particularly noteworthy fanbases.
Growing the Beard: Gaiman's editor has said that she believes issue #8 ("The Sound Of Her Wings") to be this.
Also Gaiman himself; he had a hard time figuring out the characters early on, and found the need to attach the series to the DC Universe very awkward. Issue 8 was the first time he really felt he'd gotten it right.
Alternately, issue #13 ("Men of Good Fortune") note The issue that introduced Hob Gadling could be seen as this. It marked a big step in the series breaking from its horror roots, being a largely comical story that gave humans and the Endless equal dramatic focus and featured supernatural elements as plot devices instead of as the focus of the story. It was also the first time that the series delved into historical fiction (something that it became known for), and it largely started the series' tradition of subverting and deconstructing popular fantasy tropes, featuring a notable subversion of Who Wants to Live Forever?.
I Am Not Shazam: A strange case. "The Sandman" is just one of countless names that Morpheus is known by, but he's never actually called this except for one brief instance in issue #3. The name is mostly just used to maintain a tenuous connection to the original superhero from the 1930s. Strangely, he is always called "The Sandman" in the script of each issue.
Older Than They Think: The series started as an attempt to reinvent the Golden Age superhero, the Sandman, but went on its own merry way. Its popularity led to a revival for the original hero, though (Sandman Mystery Theatre ran almost as long as the Gaiman Sandman did).
Prez was a comic book character in his own right. Right down to being countered by Boss Smiley.
Rewatch Bonus: Plenty. Most readers find that the number of characters and sidestories are really well-planned-out upon rereading the tales.
They Wasted a Perfectly Good Character: Or rather, they wasted a perfectly good opportunity for a great character in Rose's mom. Lots is made about Rose being Desire's granddaughter, but there seems to be no effect on her mother despite, you know, being Desire's daughter!. She gets basically no characterization besides her role as a mother and seems to be in no way more than a normal human being.
They Wasted A Perfectly Good Line Art: Applies to much of the art, but most especially the amazing artwork from "The Season of Mists"; Ty Bender's non-fiction "Sandman Companion" featured excerpts of the same artwork without the hideous colouring, and the difference is astonishing.
Also, Coleen Doran's art for issue 34, part 3 of the A Game of You arc, was mangled by the horrendous inking job. Fortunately, Doran had the opportunity to ink the issue herself in the Absolute Edition
In the case of the Delirium chapter, what happened was that Gaiman sent the script to Bill Sienkewicz, the artist... who apparently painted whatever it inspired him to paint. When he got the art back, Gaiman had to cut up his script and shuffle both script and art around to come with something that worked. (Frank Miller later confirmed to Gaiman that the same thing happened when he wrote Elektra: Assassin.)
The Woobie: Poor Nuala; sold out by the fairies, summarily ignored by Morpheus, dismissed peremptorily with a broken heart, treated like crap, and then the poor thing goes and accidentally helps her crush kill himself.