Gonna answer this and share my experiences without using a throwaway -- booyah!
(Source: nearly a decade of talking to/flirting with/hooking up with random girls in clubs and bars, so I can definitely say that culture difference hardly plays a part here.)
When I'm in a bar or at a club, I almost always use what PUAs call a "direct opener" -- basically, starting a conversation by expressing your interest in her. Why? Because in these types of places, there's a lot of potential distractions going on (loud music, alcohol, chatty/dancing friends, and the biggest distraction of all: other men) so you need to come in strong and differentiate yourself from every other dude who's approached her.
I always say something along the lines of "I saw you and I thought you were cute, so I had to come over and say hi." And then I reach out for a handshake and introduce myself. And the conversation begins.
There are hardly ever any circumstances in which you would want to ask for her number early on -- dude, you don't even know her, what the hell would you want her number for? You have no idea if the two of you would have anything to talk about or even get along, that's why it comes off as creepy.
Anyway, that opener buys me anywhere from 2-5 minutes worth of interaction at a busy club. There will be times when a girl just plainly isn't interested and will ignore you; but it works surprisingly well in most circumstances.
What you do next is totally up to you, but in general you definitely want to do at least a few minutes of small talk before you close for the number (or you could go and take the interaction as far as you can for the night before you close it) -- that way, you actually have something to text her about when you do text her.
While I've used that opener in clubs mostly, I've also had success using it in sit-down "inuman" places as well as coffee shops like Starbucks. It's actually better outside of club situations because in addition to the fact that nobody approaches strangers in these places, it's also a very strong and bold way to start a conversation. And I've heard that girls dig confidence ;)
One reason I think it's worked well for me these past few years is this: my mentality when using that line isn't to get anything from them. In all honesty, if the conversation ends right then and there I'm completely fine wit it. I don't even mind getting blown off and rejected (me and my friends play a game at the beginning of the night to see who gets the first rejection out of the way quicker -- it's the best way to get rid of bitch butterflies for us).
In my head, my only purpose for opening with this line is to give them a sincere compliment: if all I can do is to make her day/night a little better and that's it, then so be it. (And let's face it; whose day wouldn't get brightened up if a random stranger came up to you and told you that you're attractive and not want anything at all from you?)
There's also the indirect opener in which you start a conversation without revealing your interest (sometimes even going out of your way to downplay it). Somewhere in the replies to this post, someone mentioned "The Game" which popularized the opinion opener (i.e. "Hey, I need a female opinion on something: who lies more, men or women?" etc.).
If I have to go indirect, personally I'm a fan of using situational openers. Basically, you use the situation around you to start a conversation with them. Let's say I'm in a club and I'm buying a drink -- if a girl is waiting along with me (btw one of the best times to time to start a conversation with a girl is when she's at the bar buying a drink) a lot of times I just look at them and say "Hi, how you doing?" or if they're holding a drink already, just reach out with my bottle and say "Cheers" then start some casual small talk.
Not such a fan of this approach especially if I'm really interested in the girl though, because it takes a lot more conversation (and you have to be super interesting and/or super good looking) before she'll actually be interested enough to give you her number -- and getting the number doesn't always mean that she won't flake on you when you text her -- but it is a way to start an interaction with less risk.
But there are times when the situational opener is actually much more useful than the direct one. For example, if you're at a place where there's a lot of social accountability like a school party or whatever (friends, and friends of friends, where getting hit on by a complete stranger with a very strong come on may make you look a little too weird). Or "day game" situations like at the mall or at a bookstore or at a coffee shop where you might feel too vulnerable/exposed to use the direct opener (sidenote: it's not really a problem -- people are paying more attention to themselves and don't really give a fuck about you, so remember to tell yourself to get over yourself; you are not a unique snowflake).
Anyway, complete wall of text already so I'm going to end this here, but hope that actually helps you out.