In Gintama episode 211, it's raining when when Gintoki finds Otose stabs and gets thrashed by Jirocho.
This begins right after Naruto and Sasuke's confrontation in episode 134, as Sasuke stares down at his unconscious rival, and continues as he stumbles off to Orochimaru while Kakashi starts hauling Naruto home.
Most of the time it rains when something bad happens to the Uchiha. It rained for Itachi back when we thought he was the Big Bad and during his and Sasuke's fight.
It also rained when the Third Hokage died and when Asuma died. There'a also a perpetual Gray Rain of Depression over the Hidden Rain Village; one of the characters even compares it to the country constantly crying.
The final dramatic climax of the Fruits Basket anime occurs in the rain, after the rest of it is sunny.
Occurs in Gankutsuou. Most often in the later half of the series. Albert even walks home in the rain in a depressed manner at one point.
This happens in the 25th episode of Death Note, which has an unusually poetic kick to it overall. At a moment of high drama, it instantly becomes Dramatic Thunder.
In episode 194 of Sailor Moon, rain begins to fall when Usagi collapses in tears because she can no longer bear the strain of wondering why her Love Interest in America hasn't written back to her at all. (Unbeknownst to her, he's dead.
In Monster, Tenma sinks to his knees to curse and bemoan the cruel fate that made him the savior of a murderer in the middle of a rainstorm. It's very sad, but the water getting on his pants is like the clothing equivalent of They Wasted a Perfectly Good Sandwich.
In Robotech, one of the episodes is called (A) Rainy Night, and is just that, a rather restrained, remarkably mature (in the proper sense of the word) conversation takes place between two of the major female characters about their love lives past and present, while a soaking rain comes down outside. One of the women is involved in a very dysfunctional relationship with a mecha officer, and the other tells her the story of how she met and got together with a by-then-deceased pilot. No explosions, no gunfire, not much comic relief, the entire episode is conversation and memories of a deceased character.
Code Geass: The entire colour palette is overlaid with grey after Shirley meets Lelouch having missed her date with him because she had to identify her father's body. Whose death Lelouch was responsible for. And things just get worse from there.
In Marginal Prince, Joshua sings four times in the course of the series. Twice the songs are about his feelings and despair and of course it rains in torrents during these montages.
It's raining in the Tokyo of You're Under Arrest! when Ken Nakajima becomes the victim of the "rogue officers" that cause accidents deliberately, and breaks his leg. He's seen laying down on the rain-wet pavement and grabbing his injured leg, much to the horror of his friends.
It also rains during several early episodes of Oniisama e..., while are also those were Nanako is most bullied and harassed due to being accepted into the the Sorority. Also, it's under a heavy and cold rain that Rei waits for five hours for Fukiko, not knowing that it's one of Fukiko's cruel mind-games. It causes her to fall extremely sick.
In Sakamichi No Apollon when Kaoru finds out his father has only come to gather his stuff and won't have time for a proper visit. Kaoru thinks about the things he wants to his father about but won't get the time to, like him making new friends and wanting to know more about his Missing Mom.
Kubo generally uses rain to show when a character or group of characters (sometimes even all of Soul Society) have fallen into despair. The overriding symbolism of this belongs to Ichigo, where even his Empathic Weapon hates the rain because of the despair it represents inside Ichigo's soul. The origin of Ichigo's inner hollow lies with the proto-Arrancar White who only ever appeared when it rained and who merged with Ichigo's shinigami powers when passed down to him after hollowfying his Quincy mother. The reason why rain in Ichigo's inner world symbolises despair is because the inner hollow (as his real shinigami power) feeds on it to eliminate the source of that despair.
Second only to Ichigo, is Ryuuken. Flashbacks reveal a teenage Ryuuken was under enormous pressure to save the Quincy future. When Aizen's Hollowfication experiments bring Isshin and Masaki together, Ryuuken's life is destroyed in the process. As Masaki's designated future husband, Ryuuken's responsible for Masaki's wellbeing, but allows her to break the rules to save Isshin's life, leading to her Hollowfication. Forced to give her to the Shinigami to save her, his ability to protect the Quincy future in ruins, Ryuuken tries to walk away from the Quincies forever, stopped only by his future wife Kanae, whose fate it is to die with Masaki six years before the main storyline. Ryuuken's been living in the rain ever since.
In Manga/Saki, in a preview sequence at the end of the first season, this is present in a scene that seems to imply that a rift will eventually form between Saki and Nodoka; their backs are to each other, and one of the gifts they got for each other has been dropped in a puddle.
In one of the Dragon Ball Z movies, it was raining heavily when Future Trunks turned super saiyan for the fist time after finding the dead body of his Big Brother Mentor.
Justified in The Sandman where the Dreaming reflects Dream's emotions. After Thessaly dumps him we see him simply standing and moping in continuous rain. Mervyn complains about this causing people to have dreams of existential angst and everyone in the Dreaming to get wet, while Lucien remarks that after Nada, the entire Dreaming was razed and stayed a wasteland for hundreds of years. In an interesting twist, the landscape does not change after he is forced to kill Orpheus although he is obviously upset and grieving.
A frequent motif of Akira Kurosawa. When his protagonists are at their lowest point, you can be sure rain will follow. Totally reversed in Ikiru ;the protagonist is so happy to have achieved his final goal that not even heavy rain nor snow can stop him from rejoicing in a swing.
Parodied in Hot Shots! Part Deux with the flashback to when Ramata leaves Topper. She sends him a card to break up with him. When it sinks in, it starts to rain but entirely localised to where Topper is standing.
In The King's Speech, after an argument, Lionel tried to see Albert to apologize. At the time, Albert's brother was planning to abdicate his sovereignty to Albert, so he was told that the Duke was "too busy" and shown to the door, where it was pouring rain outside.
Used literally in Men In Black 2, as the Light of Zartha leaves Earth.
It was either parodied or used very cheesily in Alpha and Omega. When it looks like Kate has been trampled to death by cariboo, everyone is very sad and the sky turns gray and there's thunder, then Kate wakes up, and instantly the sky turns blue again without a cloud in the sky.
In a sad scene from the film Ernest Goes to Camp, Ernest lampshades the appearance of a rainstorm with a song called "Gee I'm Glad It's Raining".
In The Hustler, Eddie Felson takes Sarah out to dinner, then tells her that he will be leaving town for a little while. Fearing that he's dumping her, Sarah bolts out of the restaurant and runs straight into a downpour.
The titular character jumps from the roof of St. Bart's hospital, apparently committing suicide, and as his best friend John tries to get to him the sky goes from partly cloudy to dark and rainy in a matter of minutes.
From the same episode: as John is describing the event to his therapist, and later when he's sitting alone in 221B, probably soon after the funeral, it's again raining heavily.
This is a common trope in country music to show that the singer is depressed or there is turmoil:
David Nail's "Let it Rain" involves the singer cheating on his wife and being thrown out into the rain (apparently inspired by the last scene of the movie "The Last Kiss").
Gary Allan's "Songs About Rain" is told from the point of view of an ex-boyfriend on his love's wedding day to someone else—and as he's driving around being depressed, all the radio will play are "songs about rain."
The Eli Young Band's "When it Rains", about a bad break-up.
Rascal Flatts' "What Hurts the Most"; the singer mentions that the rain on the roof "don't bother me" but it fits with the mood of the song.
Carole King's song "Crying in the Rain" (also covered by a-ha) is about a person who's still not over a bad break-up and hopes it'll rain soon so they can cry underneath it and not let others know about their heartbreak.
Certainly at least invoked by "Cold Kentucky Rain".
"Sure Got Cold After The Rain Fell" by ZZ Top knowingly uses the country metaphor, being from Texas, although it's a blues song. However, it doubles in that the desert is cold after rain falls, what with all the wet sand and all.
Albert Hammond's "It Never Rains (In Southern California) ... it pours.
Michael Jackson's "Stranger in Moscow" about loneliness and sadness has the opening line "I was wandering in the rain", the music video features Jackson in the pouring rain.
New Radicals "I Don't Wanna Die Anymore" features "A life of rain" and "It's gonna rain" The song is about depression after the failure of a relationship. It is not completely evident if the 'two months of fun, two years of pain' in the song refers to a two month relationship breaking up and two years of depression, or about a relationship that was fun for two months and abusive for two years, though the former is far more likely. The narrator wants the girl back because otherwise he's going to kill himself.
Urban Fantasy web series Broken Saints uses this in Chapter 14, Act 2, as one of our heroes, Kamimura, walks alone through a very depressing and disturbing section of downtown Coast City—reflecting his own display of sorrow at what he sees around him.
In Everyday Heroes, when Jane is released from prison, she walks away from her family (and her old life as a villain) through a cold February rain, alone.
In Winx Club S4 episode 24 it starts raining after The Wizards of the Black Circle escape after having used the gift of life on a dead flower, taking away the Winx's only chance to save Nabu who had sacrificed himself to save the Earth fairies.
My Little Pony: Friendship Is Magic has this in several episodes, including "Magical Mystery Cure" when Twilight laments the switching of her friend's cutie marks, "Rarity Takes Manehattan" when Rarity thinks her friends have abandoned her, and "Testing, Testing, 1-2-3" when Rainbow despairs of passing the Wonderbolts history test.