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Tear Jerker: Red Dwarf
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While it's mostly played for laughs, Lister learning everyone is dead after he first comes out of stasis in the first episode was pretty sad to watch. Especially when he realizes Kochanski is gone.
The flashback in "Balance of Power" of Lister having fun with his friends, annoying Rimmer and being teased about Kochanski.
Past!Lister: All right guys, leave it alone, leave it alone, leave it...
*Fade to the present, where a forlorn-looking Lister, the last human being alive, is sitting by himself in the empty bar*
Lister: ... alone.
The end of the episode "Waiting For God" shows an old cat priest who's believed in Cloister (Actually Lister) his entire life. He then rants about how it was all a lie, lamenting all the time he wasted worshiping his god. It's made worse by Cat's complete disregard for the old man's speech. Crosses into Heartwarming when Lister does show up, though.
The scene in "Better Than Life", where all that post comes through from the past. Rimmer gets one about his dad being dead, Lister failing to figure out the scrawling writing for ages, then finally Rimmer realizes. He's been dead for millions of years but it sort of just hits him here. The next scene shows Rimmer on the observatory tower, just musing to himself. Lister comes up and actually asks if he's okay, and Rimmer goes on about how while he hated his parents, he just wished he had been able to impress them just once. The end is ruined by the Cat making a joke, but still, very well written stuff.
The oft overlooked episode "Thanks for the Memory". The scene where Rimmer, drunk to the gills, is talking about his regrets is actually quite touching.
Rimmer: "That was going to be our song... but I never found anyone to share it with. So now it's my song."
Also from that episode Rimmer's rejection of the love Lister showed him, purely because it wasn't his memory to begin with. Especially as Lister claims Rimmer ultimately felt more for Lisa than he ever did - whether or not Lister was telling the truth is never spoken of again, but it sure looked that way.
Rimmer: The woman I loved, never loved me. She loved you.
Holly in "Queeg" just before his deletion. Holly gives the crew all very nice farewell's, then he sings the song "Goodbye to Love" as his voice fades out, and we see the words "ERASE COMPLETE" on the monitor...
Lister's despondency upon learning Kryten in nearing the end of his service life in "The Last Day".
Lister playing his guitar one last time before he burns it in order to stay warm in "Marooned". Sure he still can't play, but that tone was just heartbreaking, despite the fact he decided not to go through with it a second later.
The ending of "Camille", sure it was lifted straight from Casablanca, but still.
In "Justice" when the prison A.I. scans Rimmer for guilt, and it picks up on his feelings of remorse over the accident that got the crew killed three million years ago. Sure the subject is normally played for laughs, with Rimmer all Never My Fault. But it's in this episode that we learn for the first time just how much the accident really haunts him.
Rimmer and Ace throughout "Dimension Jump", Ace tries his damned hardest to be nice to Rimmer. And as always, Rimmer just goes around making baseless accusations and barks insults.
The ending of "Meltdown", when it's revealed that Rimmer's "tactics" ended up wiping out the wax droids entirely. Lister clearly viewed the droids as being sentient, and he's distraught to hear about how they're all gone. The fact that Rimmer is oblivious to why Lister would feel that way only makes it worse.
Rimmer: I brought about peace. Peace, freedom and democracy.
Lister: Yeah, Rimmer. Right. Absolutely. Now all the corpses that litter that battlefield can just lie there safe under the knowledge that they snuffed it under a flag of peace and can now happily decompose in a land of freedom. Ya smeg head.
Lister then swallows Rimmer's light bee like last time, but it was messing with Rimmer's light bee that caused him to go nuts (well, more than usual) in the first place...nice job breaking it Lister.
The off screen execution of Winnie the Pooh, even if it was played for laugh's.
In "Holoship" when Rimmer confesses to Nirvanah he cheated on his test and explains why.
Rimmer: Every time I look in the mirror, I see this. (points at his letter H) Only to me it doesn't mean hologram, it means half-wit, hopeless, hideous failure. This was a chance to be somebody. Somebody I liked.
The shocked look on Rimmer's face when he learned Nirvanah was his opponent, and had sacrificed her place (and her existence).
Watching the alt!Rimmer and alt!Cat being vaporized by the Inquisitor near the end of his title episode.
The alternate Lister and Kryten seem to be pretty decent guys, before they get killed horribly by the Inquisitor.
Watching Lister in "Demons and Angels" being forced to murder High!Lister and High!Rimmer while under the control of the Lows.
While they were na´ve to the point of idiocy, watching all the Highs being murdered by the Lows wasn't quite as funny as the laugh track would lead one to believe.
Irony: The episode was filmed in front of a live studio audience.
High!Rimmer's demise seems the most tragic, as he at least appears to appreciate what's happening to him.
Near the end of "Back to Reality", before it's discovered they're only hallucinating Lister learns that in real life he's a mass murdering fascist. Kryten then shoots and kills a man, which obviously traumatizes him.
Near the end of "Out of Time", after the crew refuses to help there future selves upon seeing what they have become. The future crew out right viciously attack their past-selves, actually willing to kill there younger selves. When Rimmer shows uncharacteristic bravery, "Better dead than smeg!" we're led to believe we'll see a Crowning Moment of Awesome. Instead were treated to watching Lister, Cat, and Kryten all dying horribly.
Watching Starbug get blown up.
Future!Kennedy's last words in particular.
"Thank you for giving me a chance to... be reborn."
The seventh-series departure of Rimmer in "Stoke Me a Clipper" when he takes up Ace's mantle.
Ace's death, very brief, but very sad.
Ace: Well done, Arnie; you've done us proud. Smoke me a kipper. I'll be back for... (Ace's Light Bee then crackles and his image glows white before fading completely)
And Rimmer's reaction, he's just so shocked, the closest thing he's had to family in over three million years blinks out of existence right in front of him. And not just family, the closest thing his had to family who's ever been kind to him.
Cat: What's happened to goalpost head?
Rimmer: (still in shock) No, you don't understand. It's not me, it's him.
The theme that plays when Lister and Rimmer discover the space cemetery. There's so much depth there, the music encapsulates the beauty of the scene.
Lister leaving his son (himself) under the pool table in "Ouroboros" and stating that he's never going to be alone.
"Epideme", at the part where Lister decides to kill himself and the virus with him rather than allow the rest of the crew to become infected.
The loss of Starbug in the opening of "Back in the Red", even if it was a rundown rust buck held together with nothing but mold. It was home for the crew for years, and unlike "Out of Time", it's gone for good.
While sucking up to Captain Hollister, Rimmer mentions it's Hollister's anniversary and that he must be missing his wife terribly. Now take into account the fact that Mrs. Hollister's been dead for over three million years...
The ending of Series VIII ("Only the Good") might also fit the bill. Rimmer returns from the mirror universe (he actually comes ''back'') to find that the ship is burning, the others have escaped and he's stuck. Then he discovers that the antidote formula he brought back is actually the formula for the virus. And the paper burns up. With violin music. It's the quiet little "smeg" as he collapses.
Watching the ship itself falling apart.
Back To Earth
Interestingly enough, several of Lister's scenes in "Back to Earth" can potentially qualify for this. Watching Lister break into tears as he attempts to read Sense and Sensibility to Kochanski's grave shows that Doug Naylor can write excellently, a stark contrast (in the eyes of some) to the last two series of the show. It's only made more powerful by the fantastic music and the stunning (considering the shoestring budget) setting in an observation dome.
While it's played to it's fullest for laughs, Irene's death at the end of "Entangled" is actually really sad.
More or less anything explaining why Rimmer is the way he is unbearably tragic.
The flashback seen in "The Beginning" during Rimmer's time as a student at college on Io. His own father is the teacher (naturally). Rimmer's father uses his own son as the unwitting guinea pig for a class project about psychology. Rimmer's father then makes the students ostracize Arnold so he'll feel unsure of his decisions. ...Smeghead.
Considering what we learn in "The Beginning", Mr. Rimmer had been very aware from the beginning that Arnold wasn't his son. So it's highly possible all the abuse that Mr. Rimmer laid at Rimmer's feet growing up was simply doing everything he could to spite his wife and "Dungo".