In original Fallout, in Shady Sands, IIRC, people complain about not raining much lately... they irrigate their crops, but hey, it's rain.
I think there a few more people complaining here and there, but nothing specific.
Don't know about Fallout 2. I can't seem to recall, but I gather that if there was rain in Fallout 1, no matter how sparse or radioactive the raindrops would be, there was apparently enough of it for the soil to 'stabilize', as a Fallout 2 world has far more plants and green in general.
Fallout 3 - depending on whether you consider it as canon or not - was supposed to have rain implemented, I think. Weather, in general, but that would be too much of a strain on engine. I think I've read that somewhere, maybe some interview with devs, but I can't be too sure, so take it with a grain of salt. Mods fixed that, though.
Fallout New Vegas is set in Mojave, a desert, so there isn't exactly any rain there, and is mostly dry, but in Honest Hearts DLC there is actual rain falling, clean, non-radioactive, so there you go. Ingame, Zion wasn't struck by nukes, but that doesn't matter much as global climate would be affected by radioactive particles, so radioactive rain would fall everywhere. Since that's not a case in Zion, I think we can safely conclude that majority of global climate is 'purified' and that weather is getting back to normal.
Now, I've mostly focused on the aspect of rain here, but that's not the only weather in existence.
I think sandstorms have been mentioned.
In Fallout 2 Cassidy tells us that there are twister and/or tornadoes (can't tell exactly) in Texas, so that's another element.
In Fallout Tactics in the game's beginning, basically, there's a huge storm which serves as a catalyst for game's plot (I think this is the case, actually - it's been a long time since I've played Tactics, and I've never finished it, so someone correct me on this one).
I saved snowing for the end - in New Vegas Mt. Charleston is covered with snow.
There was also a nuclear winter some time after the War, but I don't think that matters much here...
Hope this helps.