This is based on opinion. Please don't list it on a work's trope example list.
Tear Jerker / Men in Black
The second movie states that K's wife left him, of course this was so K could be in the movie, however in-universe it's rather depressing when you think about it. Being an MIB kept him from the woman he loved and even with his memory erased he still was unable be with her because deep down he was still an MIB man at heart, no matter what life K chooses to lead there's a void he can't fill. Or, rather, his decades long absence changed the both of them, and she'd moved on emotionally.
K: Son, I've just been down the gullet of an interstellar cockroach, kid. That's one of a hundred memories I don't want.
Taking it further is the realization from the third film that one of those memories K doesn't want is the memory of neuralizing J to forget what happened to J's father.
As well as J's father dying after pushing K out of the way.
There's also the "Light of Zarthra" flashback from the sequel.
K: This is wrong. It was night, and it was raining...
"It rains because you're sad."
Boy, was he right.
Griffin's backstory, especially because he's such a hopeful, kindhearted person. This is a guy who lost everything, yet still came to love Earth as his home anyway. His sadness at having failed to stop Boris before is only matched by his hope that the current situation won't end the same way.
The twist at the end of the third film, that J's father was killed by Boris, and the young J never knew what happened because K neuralized him to spare him the truth. This is all witnessed by Future!J.
"Would you like to take a walk with me, James?"
"Where there's death, there will always be death."
A retroactive one because of this Hey, K, have you ever flashy-thinged me?
Boris killing his girlfriend. This was just so heartless. Thankfully, she is saved by Boris dying in 1969.
One feels far less sorry for her when one considers she was willing to break a mass murdering killer out of jail.
It's not clear that she's even his girlfriend. She could just be some girl with a bad case of hero worship or an in-film case of Draco in Leather Pants.
Part this, part Nightmare Fuel. When J time skips he is hit in the face by a newspaper on the stock markets crashing, and as he's falling other men in suits are jumping out with him to their deaths. Ouch man. Ouch.
Orion the Cat in the first film. Before his owner was murdered, he really seemed to love and spoil him.
Dee's neuralization. He just looks so heartbroken in the moments leading up to it.
The end of "The Neuralyzer Syndrome," where the neuralyzed K is "renewing his vows" by saying one last goodbye to his father.
The end of "The Inanimate Syndrome" when Agent Aileen leaves.
J: Fuel fumes really sting the eyes, huh?
The beginning of "The "J" is for James Syndrome" starts off with Jay accidentally going for a joyride in the LTD and getting neuralyzed by the MIB. The worst of it is the fact that as they do it he starts screaming in protest while some agents hold him to keep him from getting away.
The end of "The Endgame Syndrome Part, 2", where all the people of Earth learn the truth of aliens and don't freak-out in a mass panic. Only for the MIB agents to neuralize the whole planet because they don't like the idea of losing their jobs. Talk about shallow.