A.~ One of my daughters is also very shy. She is now 7 and still tends to be quite apprehensive in new situations or in large groups. I have to do a lot of reassuring and her shyness has seemed to improve over the years. I also did what you did...enrolled her in classes, joined play groups, lots of play dates and park excursions and it seemed to be futile. I finally realized that no matter how much socializing I did with her, it wasn't going to change the core of her personality...which tends to lean on the shy side.
Now I'm way more sensitive to her personality and can predict how certain situations are going to make her feel and then I'm able to give her plenty of reassuring before she's in the situation.
What I don't do is let her shyness become an excuse for not wanting to do certain things. If she wants to be in soccer, ballet, gymnastics, do piano, etc. that's fine with me (not all at the same time, but you get the idea) but she can't quit. If she makes a commitment to do something, she has to see it through and once it's over she can make the choice of signing up again or not. She's older now and gets it, but it's no different than when she was 3...I always made her see a class through to completion because I don't want her getting into the cycle of starting and quitting if it's not going her way.
I tell her all the time that I was painfully shy as a little girl, but had to make the choice to join clubs, make friends and put myself out even though it sometimes didn't feel natural because loneliness isn't a condition of shyness it's a choice and although she is shy the best remedy is making good friends and finding activities that she is totally comfortable with.
That said, yours is only 3 and won't quite get all that, but if you say it enough over the years she will eventually get it...like my daughter. She's still shy but she's got a good eye for what a friend is and a great sense of what she likes and doesn't like.
Hang in there...I know it's hard to watch, but she'll be alright. Just don't push her to be outgoing, but continue to put her in situations where she can grow (w/out quitting).
Good luck.