From the ending of the pilot, the murder of Mrs. Perkins by Beverly Barlowe. Considering that Walter Perkins was already missing and presumed dead. That leaves their son Brian all on his own.
Followed by a happy Tear Jerker when it discovered in a later episode that Walter survived the accident, though he'd still have to be punished for what he'd done.
Stark's final scene with Callister in the Season 1 episode "Right as Raynes". The boy asks him if it's possible to give a computer a soul. Stark replies with tears in his eyes that he knows it is.
Dr. Carl Carlson's apparent death near the end of "Invincible". Not to mention the fact that he injured Jack, just after telling him he was the first friend he'd ever had.
It's later confirmed that he did die due to being absorbed by the Artifact.
A brief moment in Season 1's "Dr. Nobel": While explaining nuclear fission to Zoe, a scientist draws an analogy to a family that splits up due to their daughter's delinquent behavior. This causes Zoe to briefly contemplate that she might have been responsible for her parents breaking up. It's played off as a joke a couple seconds later, but still.
In the Season 1 finale, after experiencing the altered timeline where Henry prevented Kim's death via time travel, causing a world-ending paradox, Jack is forced to travel back in time to prevent the interference. This results in a heartbreaking scene where Jack must physically stop Henry from rescuing his wife from certain death, as he screams and cries for him to move.
"Family Reunion", Jack's story about Angela Fairfield, his high school sweetheart, after Zoe digs into it. He was seventeen, it was her sixteenth birthday, and she wanted Jack to teach her how to drive. She was doing great, they were celebrating, swerved to get around another car, and something bad happened. When the police arrived, she was dead in Jack's arms, they assumed he had been driving, and he was charged with reckless endangerment. So every year, Jack sends flowers to Angela's grave, and every year her dad calls Jack to thank him.
Dr. Todd talking about his failed experiment in Try, Try Again, and how they were forced to kill the subject of it in order to save Eureka, something which clearly haunts him. The very real fear that the same thing will happen to Fargo in the present is also pretty upsetting.
Nathan's final scene in Season 3's "I Do Over" and it's aftermath. Nobody saw that coming...
The Dark Reprise of the oft-repeated scene of Jack going to Allison in her tent. The opening lines of the dialogue are the same every time, but the meanings different. And the last time, the meaning has changed completely.
While the death of Leo Weinbrenner in that same episode is easy to overlook in the light of that, he is presented as a fairly decent guy, trying to do the write thing and fix his mistake, while getting Carter's help to make sure that there'll be a way to save the situation if anything happens to him (which it does).
The secret behind Eva Thorne/Mary Perkins past, and the men how died in the bunker.
Henry's awkwardness around Kim 2.0 in "Ship Happens". Kim 2.0 doesn't even know about the original Kim's death till she's told.
Kim 2.0 requesting that she be destroyed in order to protect everyone when she believes she might be accidentally electrocuting them, like the unfortunate Dr. Lieber.
Kim 2.0's death in "Shower the People". Just when it looked like Henry had found happiness (again), it's snatched away (again!) in a very cruel way.
Jo interacting with Zane after the timeline shift. For her its basically like the man she loved, and who loved her, has died, and she has to face the ghost of him every day. Every single time she has to be near him, you can see the pain on her face.
Really, the first realization from her that things have changed. She goes to Zane, talks about their relationship and how she realizes what he means and she will marry him. Zane (who's lounging in a jail cell) just looks at her, then bursts out laughing. "Me marry you? In what universe?" He chuckles as he reads and the camera focuses on a stunned Jo realizing that she and Zane never even dated.
In "I'll Be Seeing You", when Jack finds out Allison died.
Grant's reaction when Jack tells him what happened.
"Clash of the Titans", Tiny's "death" via explosion from the Titan samples she was caring becoming unstable.
"This One Time at Space Camp..." Zane's reaction to Jo dropping out of the Astraeus program.
When S.A.R.A.H. started to melt in "One Small Step..." (It Makes Sense in Context) and told Carter that it was actually hurting her. The way she said it, in that meek little whimper. That moment really tore at my heart strings.
From "One Giant Leap..." Jo deciding to leave Eureka.
In "Lost", Allison reuniting with what she believes to be her now much older daughter Jenna.
Allison catching Jack and Jo together.
"The Real Thing", after spending the episode fighting a dragon, and the reveal that Senator Wen was behind the events of the Season 4 finale, we get a painfully slow execution of Holly, simply for being too smart. Talk about Mood Whiplash...
When Andy and Jack are in the jail cell together and Andy reflects back to when he got stuck on Titan.
Andy: There has to be a moment when you give up hope. When you forget about what's up there, and you focus on what's down here.
"Force Quit", Fargo's simple lament to Carter after learning he lied to him about Holly's death;
Fargo: You lied to me...
And immediately after that:
Carter: I was trying to protect you.
Fargo: But I couldn't protect her.
Carter: I know. Neither could I.
Both scenes in "Friendly Fire" where Fargo is going through Holly's stuff. Especially in the second scene when Parrish comforts Fargo by playing D&D with him.
Allison treating Jo rather cruelly while her mind is trapped in Jack's body in "Jack of all Trades".
Grace finally admitting to Henry that she's still having difficulty coming to grips with what happened to her in the matrix.
Near he end of "Worst Case Scenario", Holly (whose mind is stuck in the matrix), upon realizing that she's dead.
Holly: You...can't pull me out...can you?
Fargo: No...I'm so sorry.
From "Smarter Carter", Jack admitting the reason why he want's to keep his artificially enhanced intellect.
Jack: For the first time, I fit in. I don't want to go back to being the 'dumb old sheriff'.