WHEN
Obviously there are many ways to do it but I will try to describe the optimal way.
You should exchange contact information only after you have shared an experience together. This shows that she's not just another conquest but you have some connection. It also gives you something to reference when you reach out to her later.
INTENTIONS
Your intentions should not be to get the number, but rather to connect on whatever it is you are talking about. Show her a good experience now and invite her to share an even better experience with you when you can make time to meet.
Invite her to join you on an instant date that you were "going to anyway". Only when she declines do you get the number.
TIME CRUNCH
Sometimes you won't have a lot of time to share an experience together. In that case, make the contact information part of a game you play, so at least you have that experience.
"Pancake or waffles?" "Waffles, good answer."
"Red wine or white wine?" "White wine! That says a lot about you." (intrigue)
"Night in or night out?" "Sounds like a plan! Let's do it."
(credit - some TV show)
After the game tone it down and be serious. You've shared your moment, and now she should participate in the exchange of contact details. As Will Smith says, women want range. Display it.
WHAT TO AVOID
Try to avoid using the word number. The word "number" might trigger thoughts of other pickups throughout her life, depending on whether she lives in a big city / gets hit on a lot. It also subconsciously makes her feel you want something from her -- her number, and is more impersonal. The number is just a detail.
Instead say, "Why don't we stay in touch" or just make plans and say "Alright let's do it!"
WHAT NOT TO AVOID
Some people think you have to appear non needy all the time. Actually showing vulnerability when you'd like to keep in touch is good. Women have nurturing instincts and they connect with you better if you are both on the same level once in a while, instead of you always being the cocky bastard or a salesman.
If you do say "let me take your number" there will usually be a brief pause. Fill it in with an explanation such as "I would love to talk to you more but my taxi is waiting for me." Or something like that. Something that puts you on a personal level. Also this relaxes her and she is more likely to go through with it.
RESULTS
See PATTERNS in my previous post - they are a great way to get people to go through some routine with you together. You want to do the pattern that will give you the best result overall. What is that result?
You want to have her number. Also you want your number to be in HER phone with a great nickname, so that when you text or call she can remember the connection you had.
So a great pattern would be to give nicknames to each other. Except it won't be to each other. Each one of you will compose nicknames for YOURSELF in the other person's phone. Besides connecting you, it will also serve to differentiate you from everyone else in her phone book.
Right now, think of a nickname you want to give to yourself, and use it with all the girls you meet. See all the cool nicknames they come up for themselves! You can often tell what kind of girl they are by exploring this later.
HOW TO DO IT
Give her your phone and tell her that you'll need a cool nickname for her to differentiate her from every other cute girl in your phonebook (teasing). Tell her to think of one (getting her to play along). Also this takes the focus away from the super impersonal "number".
Look at what she wrote and have a good laugh together. Touch her, get closer to her, call her phone in front of her so she will take it out of her perse. You tell her to give it to you because you want to put in your own funny nickname. But she can't look until you leave. Ask her how to save your number from the call.
Save your number with your nickname, go to the front screen and hand her the phone.
You can now either continue talking or say good byes and she will try to find you in her phone later
Trust me, that is a way better connection than just "can I have your number" "oh ok thanks".
Now anytime you text or call she will be more likely to respond playfully to you! It makes a big difference.
FACEBOOK
Alternative is "do you have a facebook". It's a very low key approach and sometimes will give you more information about her!
If she tells you she has a boyfriend, I personally would not go for the phone number if I just met her. That's A) disrespectful to her boyfriend and B) she may be thinking you will call when she is out with him.
Instead, simply say "oh, do you have a facebook?" When she says "yes", take out your phone and just ask her what her name is on there. If you have a smart phone, just have her dictate her name to you and you search for what she tells you. This is something I call LEADING THROUGH FOLLOWING. Basically you tell her what is going to happen but then you act like you're just following her lead. If she is leading and calling the shots, she won't say no.
Why do I recommend facebook if boyfriend? Because this communicates *I want to be friends*. You will get to know this chick over time, and
A) you will see her relationship status and know if it ever changes
B) you can invite her out with a group of people, and get to know her and her friends
C) she may introduce you to her friends! who may be even more attractive for you than she is
Facebook helps you organize group activities to some extent. Be sure to make it clear that you are just friends, and she will be the one who will be open to something more if she ever becomes single. Let her make the decision after getting to know you. Trying to get her to cheat on her bf is being a douche. If you just be a cool guy then your reputation will bring you more girls of the type you want.
FOLLOWING UP
This should be its own post, but basically I would follow up by text or facebook the next day.
CHAT