At least one of the episodes "Ballad of Fallen Angels", "Jupiter Jazz", "Speak Like A Child", "Hard Luck Women", or "The Real Folk Blues" (and by extension the numbers "Space Lion" and "Blue") evokes very powerful, bittersweet emotions that are enough to make most cry.
Specifically in "Speak Like A Child"... "I can't remember".
Spike's last stand and possible death.
"You're gonna carry that weight..."
The last shot of Spike's face, then those words are so powerful.
Even earlier than that, Faye confronting Spike before he leaves and her consequentially breaking down when he brushes past her completely tore a few viewers apart. And dammit it's the last time we get to see Faye too!
Also his "look at my eyes" speech.
When Julia is shot and killed. You might see it coming with how she awkwardly got up, but it was still so sudden, then the silence, the birds, the white. What can really start the tears flowing was Spike's face. 26 episodes we watched Spike do it all with this calm and way to cool look on his face. Then we see his look of utter shock.
On "Pierrot le Fou" on Youtube almost every single comment lampshades the trope: they're all about how horribly sad the ending was and how NO viewer could stop the tears when Pierrot, in his Villainous Breakdown, screams and cries like a little kid after Spike throws a knife at his leg.
Or the closing moments of "Waltz for Venus."
"Father-person gone." Yes, Ed has a tearjerking moment.
The ending of "Hard Luck Woman" deserves special mention: after spending the entire series with no memory and only a few teasing hints about her past, Faye finally remembers who she is and where she came from. As she's preparing to leave the Bebop - smiling and genuinely happy - tells Ed that having a place to belong is the best thing there is. When she gets to her old home, she finds it abandoned and destroyed, with only the foundation remaining. As "Call Me Call Me" plays, we see Faye using a stick to draw the outlines of where the furniture used to be in her old room, and then lie down on the spot where her bed used to stand, staring up at the sky.
It's made sadder by the shot of Spike and Jet in the ending montage of "Hard Luck Woman," after Ed and Faye leave. Having set out dishes of hard-boiled eggs for dinner, Jet and Spike sit in total silence, cramming the eggs into their faces. When they finish their own bowls, they start in on Faye's and Ed's, which doesn't seem very sad — until you realize that they're avoiding talking about their friends' departures. The next episode, they're moodily talking about how their friends were nothing but trouble, and how it's good that it's just the two of them again. Their words fool no-one.
The quickness of Ed and Ein's departure. She doesn't stick around to say goodbye, she doesn't make a big show of it; she just draws her enormous signature smiley face on the deck of the Bebop and takes off. The scene with Jet and Spike surveying the picture and realizing what it means is enormously touching. Add to that the fact that Faye eventually comes back and Ed doesn't...
This video sums up some of the tearjerker moments quite well.
Possibly one of the most tear jerking moments is the very end of the episode, when we see what Spike did with the paper-pinwheel that Ed gave him. Spike, the seemingly-callous, lackadaisical child-hater, took the paper-pinwheel, and taped it to the bow of the Bebop. Manly Tears.
A special mention has to go to the scene where an old lady, who was a classmate, remembers Faye and is mystified by her young looks. She attempts to engage her in conversation, but Faye simply says she's a ghost and runs off. Seeing what Faye should have been and what lives she missed out on (and the people that missed her) was just brutal.
From the very end of the episode: "See you cowgirl, someday, somewhere!"
The ending of "Brain Scratch", already one of the darker episodes of the show, gets a bit depressing once you start realizing just who the 'villain' was and what motivated him, despite what he did to all those people. And then there's the background song used repeatedly through the episode, "23 Hanashi", that we finally get to hear all through at the end of the episode when Jet pulls his plug and isolates him from the rest of the world. The angelic choir ending really punches it in.
And Ed's simple, innocent wish that the boy will have good dreams now.
"Sympathy for the Devil": Sure Wen would take catatonic men to use as a parent and seemed to kill without hesitation, but consider that he has been alive for nearly a century trapped in a child's body, immortal and seemingly unkillable. The tear jerker comes from when we see how it happened, with Wen surviving the moon/gateway explosion and having been shielded from the blast by the (now charred) bodies of his parents. When we see the look on his face its pretty obvious that mentally he is no longer a child anymore.
Two of the most moving moments in Cowboy Bebop come simply from the music being played over the credits: Space Lion in "Jupiter Jazz" and, of course, Blue in "The Real Folk Blues" are sure to wring a few heavy emotions out of you.
"Ballad of Fallen Angels". Three of the most riveting and truly tear jerking moments came from three songs. "Ave Maria" "In the Rain" and the legendary haunting melody choir "Green Bird". This episode is just one out of many emotionally driven episodes.
"See you, space cowboy." The finality of that statement just hangs over your head.
"We'll never get out of here now... I'll never see Mars..." *BANG*
Faye and the rest of the crew watching Faye's time capsule message to herself. The reassuring words do almost nothing to improve the depression in her eyes.