I'm not sure if this applies to all girls but, the best way to approach me is to walk up to me and say hi. I'm very shy, but pretty friendly so if you initiate a conversation, I'll talk to you. Normally when I try to initiate conversatitions with people, it just sort of dies and get awkward, so I've shy'd away from approaching others, even though I shouldn't.
I don't really like the direct method, because I need to know someone first, before I can feel comfortable going out on a date with them. I realize you can get to know someone on the date, but often people don't really act like themselves on dates, or I may feel too much pressure to impress.
Plus, if a guy just approaches me, without knowing anything about me, I feel this is too look-based. I went to a friend's birthday party at a club, and when you go to the club, you generally are looking your best. So when this guy at the party was all "oh you're so hot, you should give me your phone number" I felt weird. I'm not a person who is confident in my looks, so then I'd feel I'd always have to be keeping up apperances. What if he saw me without makeup and thought I was ugly? Why is he only basing his judgement on looks? I could be the meanest girl in the world, or a total weirdo...
Another good method to meet girls is through mutual friends. Seriously. For some it might be weird to date a friend's friend, but seriously the opening is given to you. Just ask them to hangout, or get your friend to invite them out with you. If your friend was a good friend they'd be wing-man(or woman) to you. lol.
I think maybe girls don't ask out guys because it's an ego thing? You never know if a guy is going to feel that it's a blow to their masculinity or get a big head. Whereas, girls expect to get asked out so it's not a blow to their feminine nature or whatever, and often (but not always) they won't be big-headed about it. Most girls have a body image issue, so they don't take rejection well, seeing it as a personal blow. Or they may see it as not "feminine", or that it might be interpreted in a bad way. What many girls don't realize is that guys also have self-esteem problems. Men are supposed to be the "confident" ones, so girls don't expect them to experience anxiety over asking girls out or whatever. So they think men will have an easier time asking out girls, or dealing with rejection, even if that's not the case. Uh, that's just what I think though...