In Back to the Future Part 2, it starts raining soon after lightning strikes the time machine and Marty is left stranded in the past again. Justified in that this is the same storm that played a role in sending Marty back to the future in the first film, and there were all sorts of signs of a storm brewing the whole time anyway.
Happens in a scene of the French comedy Bienvenue Chez Les Ch'tis. In France, the northern regions are caricatured as cold and ugly; the main character of this film is transferred there as a sanction for trying to fake a handicap for a job. He prepares for hell, and just as he crosses the sign of the dreaded region, well, this happens.
The Lord of the Rings plays this trope in The Two Towers, when they're about to be sieged by an army of about ten thousand orcs, with an army of about five hundred, many of whom are elderly men or extremely young boys:
Gimli (to Aragorn): Well, lad. Whatever luck you live by, let's hope it lasts the night.
In Sanford and Son the pair have a disastrous camping trip and Lamont notes that it could be worse. When his father asks how, Lamont notes "It could snow." Sure enough, it instantly starts snowing right there and then.
Turned up an extra notch in That's So Raven: Raven got stuck in a limo's moon roof when going to prom. Then it started raining on her. "Well at least it can't get any worse"...cue the gale force winds.
Inverted in The West Wing, where rain was seen as a good thing that needed to happen in a string of good things happening—it turned away late voters who would have voted for the other guy, and the rain was literally called on cue by Will (with a lampshade with a "I didn't know I could do that"). It would have been played straight if it was from other guy's POV.
Also inverted in How I Met Your Mother, where Ted calls down a deluge in a massive thunderclap by screaming "COME ON!!" at the heavens after doing a desperate Rain Dance all afternoon. However, it played this trope straight for Sandy Rivers, who was going to get laid on a planned camping trip that was canceled by the rain. In fact, Sandy is seen crying while he reports on the downpour on TV.
The Russian punk rock band Sektor Gaza has a song "Tuman" ("Fog") with lyrics that literally translate as: "We're getting the jitters / The rain started suddenly / The stiff, pouring rain will finally kill us..."
The lyrics to Shampoo's "Trouble":
We couldn't get a cab, 'cause we ain't got no money! We missed the last train, but we thought... We'd get the night bus, but the night bus never came! We'd almost gone home, and it started to... * thunderclap*
Everybody out of the Water by The Wallflowers starts with some catastrophe followed by rain:
Now on your marks, get set, let's go. You gotta move now before she explodes. Look out the window, look down below. Back away from the glass! There she blows. The city's been leveled, the hills are in flames. The streets cracked open and they're pushing up clay. The temperature's dropped. The sky is gray. And it ain't even over. Here comes driving rain.
Calvin's dad: At least it's not snowing! Right? Right? (Later, as they sit in the rain eating cold ravioli out of cans.) I mean, say it was snowing so hard we couldn't make a fire.
Naturally, the rain stops the exact moment they decide to leave.
Company, "Getting Married Today," after the second verse of Amy's nervous breakdown:
Bless this bride, Totally insane, Slipping down the drain, And bless this day in our hearts (thunder) As it starts to rain.
In Chapter 1 of Mother 3, after Flint and Fuel get out of the Sunshine Forest, Bronson comments that the only way that the situation could get any stranger is if it suddenly started to rain. It does. Unusually for this trope, however, the rain is a good thing, as said forest was on fire, and the rain put out the fire.
Subverted in an episode where the Simpsons lose their house. Tossed out unto the street, Homer says, "Well, it could be worse. At least it's not raining." (Beat) "See? Told you it could be worse."
In another episode, Mr. Burns is telling the story of how he went to jail. As Smithers leaves, Burns notes that this the point in a story where it would start raining, and decides that, since he's telling the story, it did rain. Then he decides that rain wasn't depressing enough, so he has it snow instead, capping it off with Smithers losing his nose to frostbite.
In the Family Guy episode "Brian Writes a Bestseller", Stewie is stuck without a ride and without cab fare. He says, "Well, at least it's not raining." Promptly subverted when a man runs up, stabs him, and leaves him for dead.
This joke gets a call back in the episode, "Stewie Goes For a Drive", where Stewie says, "At least I'm not getting stabbed by some random guy on the street!", and it starts to rain. He then tells the viewers it's the exact opposite of what they were expecting. And then the random guy shows up and stabs Stewie anyway.