This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Boardwalk Empire
Jimmy hitting the opium den in the ending sequence of "Nights in Ballygran", where the lyrics to "Carrickfergus" really sets the tone and gives the viewer a fair impression of what he must be feeling after Pearl's suicide.
Nucky's revelation about his son's death and his wife's suicide.
Richard's scrapbooking in "21" and his suicide attempt in "Gimcrack and Bunkum".
Richard's confession to Angela that when he returned from the war with his injuries and his twin sister nursed him back to health despite them, he felt nothing to her.
Nucky at his father's funeral; he's The Stoic upon hearing the news of his death and flatly refuses to join Eli in sentimentalizing him; then, after Eli leaves, Nucky notices his father's shoelace is untied in the casket, reaches in to tie it, and breaks down crying.
The end of "Georgia Peaches": Angela sobbing over the body of her lover, pleading for her life with Manny Horvitz, and Manny telling her that Jimmy did this to her before shooting her in cold blood.
"Jimmy, I have to leave now. I'm sorry."
All of the flashbacks with Angela, when you see how young and innocent they were.
"To the Lost" is one big one, in retrospect - Jimmy has been tying up loose ends, making sure his son is provided for, apologizing to Nucky, standing up to his mother, making sure Richard is all right... before going on a suicide mission.
That not only Jimmy, but Richard, knows what's coming makes the episode even sadder.
Say what you like about Gillian, but her letter to Jimmy in "You'd Be Surprised" is heartbreaking on a number of levels. Not only can she not acknowledge that her son is really gone, but her loneliness has almost gotten too much to bear.
Similarly in "Ging Gang Goolie" when she's putting away all the photographs of Jimmy, you know just how much she loved her son (in her own way).
Margaret upon discovering Owen's body. Made even worse by The Stinger - she's pregnant with his baby.
Nucky finding Gillian in the hallway drugged out after getting an overdose of heroin by Rossetti, reverted to her 12 year old self telling him she "was good" and the the man brought her upstairs and "did something bad". Reminding Nucky (and the audience) how she got so twisted in the first place and who's responsible.
Gillian starts the season in pure Alas, Poor Villain territory. She's now addicted to heroin after Gyp forcibly injected her, and spends her days completely alone except for the johns she picks up to scrape by, under pretense of selling them the house.
After Richard returned home, his family beloved dog is dying and Emma asks him to Mercy Kill him. However, Richard, having grown weary of violence, find himself unable to do it. As Emma quickly takes the gun herself and does the deed, Richard tell her that he "doesn't want anymore of it".
After Eddie was arrested by Agent Knox, during the interrogation his entire back story of cheating on his wife and stealing from his company was laid bare. He returns to Nucky's house, and while Nucky can't be blamed for having a family crisis come up all of a sudden, he should have known better than to simply assume Eddie's weak cover story was true and complain about a relatively minor problem of mismatched socks. Owing to this series of events, feeling entirely alone and with nobody to help him, he writes a suicide note in German, then jumps out the window.
Gillian being betrayed and having her world shattered by a man again, then being "crucified" by three men holding her on the ground when she attempts to flee.
After a botched assassination attempt on Narcisse, Richard is mortally wounded and dies while hiding under the boardwalk. His dying dream is being reunited with Emma, Tommy and Julia, with his original, unscarred face.
Chalky's reaction to his daughter's death. Damn Williams deserves at least an emmy nomination for it
"Devil You Know" is one for Chalky in particular the scenes where he begins to tear up after hearing Daughter's singing voice again and when he goes peacefully to his death, remembering her voice. When he dies, the music stops, and during the credits, instead of the usual song, all we hear is the soft scratching of a finished record.
In the same episode Eli's emotional reaction Van Alden's death.
The finale has several:
Nucky and Margaret's last dance.
Al Capone's farewell to his son as he goes to prison. "Just remember, I did it all for you. That can't be for nothing."
Nucky selling Gillian and his soul to get ahead. The worst part is when she gives him this gaze of absolute trust and you know she has no idea what the Commodore is going to want with her.
Gillian: Mrs. Thompson said you want to be good. But you don't know how.
Nucky getting shot to death in public by Tommy Darmody. Even if one doesn't feel sorry for Nucky, it's still tragic, because Tommy is ruining his life for an act of pointless revenge, making all of Richard's sacrifices for him worthless.
Consider Tommy's words to Nucky, "That's your answer to everything." And then remember Nucky's last visit with Gillian.
One can only imagine how heart-breaking it is for Julia, who has been Tommy's surrogate mother all this time.