This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / American Gods
Some of the interludes are really depressing:
The story of Hinzelmann, the child in the dark
The story of Wututu and Agasu. Even the lead-up to their story proper is tough to sit through as it includes a thought exercise about a Nazi commander in charge of a death camp to prove a point about how everyone distances themselves from the excessive pain of others.
Salim and the jinn. It's hard to decide which is more pitiful, the down on his luck man trying to sell shoddy merchandise in a city ignoring him or the ifrit, forgotten and unrecognized, stuck as a taxi driver, having to drive with way too little rest, with people not even always paying him and at least one shitting on the upholstery. And to make it worse, not too much later, after Bilqis dies, one of the first casualties in the war is an Arab taxi driver in New York, implied to have been Salim who took the jinn's place/was turned into an ifrit by him.
Atsula having to die right at the edge of the new land she is leading her people... and as she was the last shaman of her tribe, their god is one of the first truly forgotten ones.
In a book where Odin and Loki appear, surely you expect Thor to show up, right? Nope. He's dead, shot himself in the head in the 1930s, according to Wednesday at least.
It's sobering that Laura died because she cheated on Shadow with his best friend, and was giving him a blowjob while she was drunk and he was the designated driver. It's even worse when Laura starts to look out for Shadow, while resenting that he brought her back by accident.
Czernobog and Bielobog's situation. When they were children, people took one look at them and said, "this one is blonde so he must be good, this one has black hair so he must be evil." Cut to thousands of years and miles later, and Czernobog is living on pension from his slaughterhouse job which he took to sustain himself with some sort of sacrifice even if not human, without becoming a mass murderer.
Mad Sweeney's last talk with Shadow, his death by a combination of hypothermia, alcohol and giving Shadow a much more powerful coin than he should have (a coin which Shadow cannot return to him as he gave it to Laura long ago by this point) and his vigil.
When we meet her, Easter/Eostre/Eostr seems to be pretty well-adjusted and happy, saying that even today, people celebrate her feast with eggs and rabbit in her name. Then while Mr. Wednesday is taking her out to a café to convince her, he nonchalantly asks the baristas and waitresses who knows where the word "Easter" comes from. Two think it's from Latin, one says she doesn't do Christian stuff as she is pagan. When Wednesday asks the self-professed pagan who she prays and sacrifices to, she cannot give a name. When she leaves, Mr. Wednesday says that they can repeat this experience with a hundred random people (this being in San Francisco, Wednesday says Easter has good chances) and for every 10 that knows the answer to the same question, he will allow Easter to cut off a finger of his, but for every 20 that doesn't, Easter would have to spend a night with him. He adds it would be useless, because he would end up with ten fingers and with 5 nights with Easter. He even says that they might mouth the name and complete the ceremonies but they hold no meaning to them, and that Easter is just as forgotten and ignored as anyone else. Easter is visibly in tears.
After said café experience, Mr. Wednesday, hustler that he is, tries to shortchange the self professed pagan waitress with 10 dollars, but Shadow gives her that amount of money. When Shadow asks Wednesday why he did it when she did nothing wrong, Wednesday first proves him to the contrary saying the waitress killed a cat on purpose as a child to know what burying it would be like, cheated on her boyfriend and gave him an STD, stole from her grandmother and other old people when visiting the nursing home, and has been stealing from every single workplace that she had. The second thing Wednesday says to justify his behaviour is that taking where and what little he can is the only way he and his ilk can get by these days because they are forgotten and nobody sacrifices to them anymore.
When Shadow and Laura are finally reunited, Shadow after coming back from the dead and Laura ... well, as well, note somewhat. It's not really clarified if it was permanent. But she'd stabbed what became the spear of Odin through her stomach to get to Mr. World. and Shadow asks her if he can bring her back. She tells him she'd rather go the other way. He does so, with a coin trick.
Shadow finding Allison's body. He can see the fear in her wide open eyes and the tears still frozen to her cheeks.
The realization that hits Shadow and the reader: that Mr. Wednesday was his father and used him as a sacrifice to start the war between the old and new gods. Shadow doesn't even have good catharsis because when he meets the original avatar of Odin in Iceland, that Odin isn't the conniving Mr. Wednesday.
Also the fact that Shadow is an avatar of Baldur, and that not every myth of his death involve a sacrifice. Some myths have Loki getting Baldur killed by mistletoe deliberately, and Odin going Papa Wolf by asking for Hel to deliver his son back to the living realm. Mr. Wednesday and Low-Key Lyesmith could have chosen a different two-man con ...
He may have been a bastard, but that one glimpse into a far-off memory of Wednesday, showing him as a cloaked and hat-wearing wanderer all alone at Christmastime, watching a happy family through a window... Especially since Shadow also sees a similar scene just with halfway-decaying Laura on the outside and her family on the inside, with the narration pointing out that not even her breath fogs up the glass.
A pathetic and sometimes laughable guy also on the villains' side, yes, but Technical Boy's freakout at the Hotel at the Center of America, with no real electricity, no wifi, no real LAN, and no McDonald's within 30 or 50 miles.
Monarch of the Glen
Shadow's Heroic B.S.O.D. following the novel, where he's Walking the Earth and doing his thing. He doesn't even raise an eyebrow when Smith and Alice offer him a job as a bouncer for the weekend.
How Jennie describes monsters to Shadow: you know you've met a monster when you're dinner. She is a troll from the old days, and Baldur is a Norse god considered a monster by Smith and Alice.
Also her story of a huldra woman, who, like many of her species, fell for a human guy, but this one was abusive, beating her, yelling at her, drinking. After 10 or 5 years, one morning, when the huldra is making fire, the drunk or hungover man staggers in angrily demanding where the breakfast is. The huldra picks up the poker and bends it into a perfect circle without a sweat, the implication being that she could have done the same to the man's spine anytime in the past years. "But tell me Mr. Shadow, if that is the case, why did she even let him treat her like that in the first place?"
Grendel's mother starts wiping the blood off her son's face after Shadow beats him and then returns Grendel to her. She says with grim satisfaction that at least this time her son is alive.
To thank Shadow for not killing her son, Grendel's mother tells Smith and Alice that if any harm comes to Shadow, she threatens to hurt them. This is sad because Shadow still had to fight Grendel, though he didn't want to be a monster.
How Moira and Ollie met: they cheated on their partners at the time.
Moira and Ollie's reactions when Shadow tells them that Cassie asked him to say hi to them. Moira then reveals that Cassie was her girlfriend, while Ollie is disturbed enough to check that Cassie's where he buried her alive.
The Reveal that Cassie was Dead All Along and she wanted Shadow to avenge her because he was the only one who could see her. Also that Ollie murdered her even though he had already won over Moira.
Cassie's apology to Shadow for using him to avenge her and says that he must hate her. He tells her that he doesn't as she fades to the afterlife, and says he wish he could have met her while she was alive.