Make a comment on something they are looking at or wearing.
If it's at a coffee shop and she's sitting down, say something briefly to her and walk away. Personally, it's not something I've done (I'm pathetically shy myself) but have been the the receiving end of.
If it's not a good fit, it might be awkward for a moment, but then you can continue on your merry way.
Outside of TV and movies, most people you meet in public are and generally not in a social mood.
I'd recommend you save your energy and efforts for other methods with more than a 1 % chance of yielding results.
seeing as there are beautiful woman everywhere but due to fear of rejection and genuine feelings of extreme awkwardness when attempting to do something like this I remain paralyzed.
Practice just approaching strangers and talking to them.
If she continues to look at you; chances are good and you can make your approach. Generally, I absolutely love it if someone goes out of their way to start a conversation with me.
Anything, even just as simple as talking about the weather can turn into a nice chat if both people are engaged.
If she's not a total bitch, the first thing she thinks will not be, "Get lost, creep." I have never been approached in a coffee shop, bookstore, bar, or on the street by a guy ... so I doubt that guys don't find me at least mildly pleasant to look at.
I make the effort to smile at most people most of the time, so I doubt that I am completely unapproachable.
I guess this is partially why I'm attempting to meet people. If you are not so great at doing this, write your name and number on a piece of paper and quickly introduce yourself, explain that you need to be somewhere urgently and leave.
Meetup groups are something I've been looking into but between work and other responsibilities it's hard to find the time. Otherwise, just talk to her/him Honestly, as a reasonably attractive fairly normal female, I feel that I need to express the feelings of myself and many similar women I know. I'm a retired classical ballet dancer who does martial arts ...
Although it's true that opening your mouth and saying something absolutely sets you up for the possibility of rejection, it also sets you up for the possibility of a fun conversation. I remember this first dawned on me when I was single, and I was at a restaurant with a friend.