Check out the book "Technical Theater for Non-Technical People" by Drew Campbell. There are pretty specific instructions on how to do it.
Outside of rrigging a pump and pipe system and shooting water through it the way SINGIN' IN THE RAIN is done usually, I have also seen it done very effectively with rotating gobos and glitter sprinkle. The gobos are slats of light that create a moving light as if rain is falling when they rotate. Add to that either people in the flies or cans with sifters that sprinkle glitter and the light from the gobos will reflect off the glitter and create a rather magical rain. When this is done in SINGIN' IN THE RAIN it is the end of the first act, so you can easily sweep up the glitter during intermission.
That is a great idea, but how do you quickly deal with the glitter on the costumes?
With SINGIN IN THE RAIN the only actor on stage is Don Lockwood and a few chorus people run by. Then it is intermission and those few actors simply don't wear those costumes again, besides the chorus people making crosses are using umbrellas and are protected from a lot of glitter. Also, there doesn't need to be a ton of glitter, just a light sprinkle to make the lighting effect have some sparkle. If you are doing a show where the rain hits in the middle of an act, then maybe you just have to drop the glitter idea and the lighting effect is enough. This is the case in WICKED where they use a projected rain effect at one point.
That seems like a very detailed explanation of how to do it, if you want to take the time to look through it.
Now that is definitely a link worth keeping! Thank you so very much!