How do you ask a woman out on a date?

A big reason why I try to avoid direct eye contact because I know there's a good chance it will get me in trouble. Especially with my 'magnetic eye stare' which I sometimes forget about before it's too late.

I mean, it's completely understandable that a woman would find being stared at to be uncomfortable.

That's the question. I think they must mean something like 'staring with intent'. All very Orwellian but apparently all necessary to ensure that women are made to feel safe.

What would be classed as "the wrong way"?

Something like 'hey I know you're busy now but some other day would you like to go out for a drink or coffee?'

If she says no you just respondwith something like 'ok, well thanks anyway'. if she says yes have something to write your phone number on quickly so you can do that, hand it to her and say 'call me.'

Yes, I occasionally look back to school (sixth-form) when a girl I liked gave me a message asking if I'd like to go out, or something like that. So not so much the curse of subtle signs but more so how too obvious it was and so I didn't feel I could take it seriously. From one extreme to another - too subtle to too obvious - the impossible reading of implied signs continues. Not that I got any closer to getting a date from that point. Though a girl did kiss me when I was about 15, I think, in a joking way. I guess that counts as losing my virginity right?

Yes, I agree that there is a very good possibility that it could be 'flirting'. But unfortunately the world is getting more dangerous for men to respond 'in like fashion' and flirting back, even if all the signs seems to point that way. I know at railway stations there are posters saying that men could be apprehended by the police for even looking at a woman the wrong way and causing her to feel discomfort. So even more reason why I keep my head low and maybe try to catch an eye if I'm feeling foolhardy enough.

But with the example given, I'd reiterate what's been shared on here. Exchange like for like: 'nice coat, nice hair'. That sounds good. If she's interested she'll probably try to make small talk so persevere. One step at a time. Good luck!

To be honest I'm not sure if she is flirting or not,  She might just like your coat ;-) 

Things like this make me look back at my life and think of all the missed opportunities I have had with women. 

I was so naive I used to think women just wanted to be friends with me when in reality they would want to date.  I would be friends for months with them and eventually some women would eventually tell me straight to my face,  (because i missed so many obvious signals) lol

You would need to build up a considerable amount of rapport, before suggesting a date. From the woman's point of view you are an unknown quantity and possibly dangerous. Try talking to her much more before asking her to a date, when you talk to her combine an interest in her, asking about her interests, with giving out some information about yourself, especially information that shows that you are not a potential threat. If you find a common interest, say in cinema, this might give you an opening for suggesting a date. Start small, as has been suggested a day time cup of coffee invitation is much less likely to be turned down than an evening occasion.

If in the UK, maybe swap in Moonchild's 'like to go for a coffee' instead of my 'like to go on a date' ('date' is very American, I think,)

Alright, I might give that a go! 

That's really good advice, thank you. The emphasis on awkwardness seems a good way to approach things as indeed it could be awkward since it's somewhere I shop a few times a week so I would probably see her again.

Perhaps you could maybe ask her if she'd like to go for a coffee with you sometime. 

There are three possible things going on here. Here they are in order of likeliness (sorry)

1. They are a friendly person who likes your coat.
2. They are flirting, but in a friendly way and don't want it to be anything more than that (yes, this is a thing!)
3. They are flirting, and want to get to know you more.

But your question is "how do you ask a woman out on a date?"

So here's my answer...

Say, "I know that this is a long shot, and I feel a bit awkward about this, and I promise that I will never bother you again, but... would you like to go on a date?"

Be prepared for the worst, and hope for the best.

If they do not say yes, then apologise for making things awkward and wish them a good day and leave. DO NOT TRY TO CONVINCE THEM OTHERWISE!!!! Then go off an lick your wounds like millions of us have done (autistic or not) since time immemorial.

If they say anything other than yes, they have said no (this is hard for autistics to spot.)