My tried and test online formula (over many 100's of conversations) goes line this ..

First message:
Quote something interesting/funny in her profile in the message title.
Open message with some funny comment/banter about what you quoted.
Ask her an easy question, interview-style questions work just fine at this point. Don't ask tricky questions or any bullshit like "if you could be animal what one would it be" etc your trying to make this as easy as possible for her to continue the conversation.

Second message:
You need to pace it so that conversation covers a couple of days in order for to her feel comfortable so don't reply the same day but the day after you get her message back.

She probably asked you a question back, quite often "and what about you ?" throwing the same question back at you. Answer it, again no particular trickery needed. Maybe add in a bit of random banter then end the message with "Anyway, we should talk properly, my number is xxxxxx". Don't make it a question and don't ask directly for hers - I cant stress this enough. This will nicely tiptoe round any giving-number-out-to-strangers unconscious reaction she may have.

Consider delaying the give-her-your-number ploy for an extra message exchange (or even two) if you sense she needs more comfort building time like maybe she lets on shes never met anyone of an internet site or something like that. As above stick to one message exchange a day.

Almost without fail she will send you hers back or sms you or even call you. Chat for few minutes about any old shite, again no super duper "lines" or tricks needed, then casual close the conversation with "so .. fancy meeting up some time ?". Note I did make it question this time as it just seems more natural and basically you wouldnt have got this far unless she was up for meeting.

The same structure is a good starting point for online/instant chat as well - a couple of short chat sessions on consecutive days culminating in the give-her-your-number ploy but there are more variable and you have to play it by ear more - you might even be able build comfort and get the number in one session.
One thing that helps "oil the wheels" with an online chat is some sexual reference in your profile just something in there like "oh and of course I'm great in bed" etc this will give her opening to escalate the chat session to a sexual level. IME you should always let a woman make the first sexual reference in a chat session but then you pounce on it and escalate it. Once this has happened then a meeting and/or shag should be forth coming, possible that night if your really smooth