And before you ask, yes I’m gonna finish this sammich.
Not impolite to refuse.
Very, very weird to ask.
It depends on the sizes of the two individuals, and the size of the umbrella. If the person is morbidly obese, and I have a standard-size umbrella, then, sorry no. But if the person is half the width that I am, and I’ve got one of those huge umbrellas, then I’d be a dick to refuse . . . or I would offer without being asked.
And it doesn’t depend on anyone’s gender.
Good way to get pickpocketed. Absolutely I would refuse to have some stranger that close to me for a sustained length of time.
Anyway, I say why should one help out someone that can’t be bothered to carry a small folding umbrella, and expects others to put themselves in an awkward position to help you.
I’d be amused if someone asked, and I’d definitely share. I’d probably have offered before they’d have a chance to ask. I don’t think it’s rude to say no, it is your own umbrella after all. But it’s fun to be gallant and kind, I’d think them a boring stick-in-the-mud more than anything else.
I met one of my best friends when he shared his very large umbrella with me. 
Neither asking nor refusing is necessarily impolite IMHO. If you’ve got a small umbrella you can’t share; if you’ve got a big one then it is polite to share.
This reminds me, I need to go and buy a decent umbrella.
I have offered on various occasions if I’ve got a big enough umbrella that we don’t have to stand cheek by jowl to fit. I’ve never asked, and any rudeness inherent in refusing depends on how you do it.
I did once have a lady dive under my umbrella once while I was walking down the street. She just appeared right at my side (and it wasn’t a terribly big umbrella) so abruptly that I jumped back, startled. Before I could figure out what was going on or say anything she cursed me for being selfish and ran off. Strange woman.
You never know who you might meet! 
At uni, my house had an enormous umbrella, it was like a tent, it even had several layers. We could fit about 5 under it. Good time…
Impolite to ask, not impolite to refuse. It would have been polite, but in no way expected, for them to offer to share, if it was a big enough umbrella.
I could probably imagine a contrived scenario where it would be rude to refuse (e.g. rain is coming down in buckets and I have a huge golf umbrella and there’s no other shelter within a thousand yards and the other person is carrying something fragile, etc.). But in general I’m a believer in BYOU.
I might offer it to someone. But I would find it odd if someone asked.
You know the saying “Bad planning on your part doesn’t constitute an emergency on my part”?
Yea, I would not allow you under my umbrella either. In fact I would probably move to the other side of the bus stop because you’re a weirdo.
I voted no, but if someone asked me I would happily share with them.
I said “yes”. If you have space for another person, you shouldn’t refuse.
I can’t imagine saying “no”. Call me a candy-ass or whatever, but I don’t like to let people down like this.
I generally don’t follow or agree with this, “Bad planning on your part doesn’t constitute an emergency on my part,” shit.
Why does this thread remind me of the movie theater seat trainwreck thread?
Why are you so afraid of capitalization?
Nitpicking aside, because it’s uncomfortable. I mean, if I am going home, it’s fine because I can just change my clothes, but if I am going out, I can’t.
But usually, if it’s raining, I have my little fairly water-resistant coat that I wear so it usually isn’t an issue for me.
I never carry an umbrella and it has never occured to me to ask a stranger to share.
If a woman asked I would feel obliged to give her my umbrella, and then bitch about how rude she was later to my friends. I wouldn’t try to fit me and a stranger under one umbrella.
If a man asked me to share it would have to be an unusual situation else I would probably laugh. Maybe if a freak rain squall caught a clown in full makeup while I happened to be standing there with an umbrella. I’d probably still laugh.
The most important question here is, of course: Are you cute?
Seriously, I’m not letting the Elephant Man or some random junkie share my umbrella. I’m more likely to poke them in the eye with it. But a nice-looking 20-something of the female persuasion? Sure thing.*
So I’m an asshole. Sue me.
(*Not that such a person would ever ask, of course. I’m no George Clooney myself, to put it mildly.)
Does this also apply if you have space in your car and you see people standing at a bus stop in the rain? What about if you have twenty bucks in your pocket and someone asks you for five (maybe to buy an umbrella)?
I’m generally a giving sort of person with a fair amount of empathy. However, I also have an acute sense of personal space. If someone asked to share my umbrella, I’d probably just give it to them and stand in the rain myself rather than willingly stand that close to a stranger for an unlimited amount of time.
So, no, I don’t think it’s rude to refuse, and it’s at the least a little weird to ask if it’s not offered.
You’d be putting yourself at risk of getting sprayed with seltzer water.
Unless you’re Sofia Vergara, you wont be sharing my umbrella.