In America there is nothing to talk about. Except for Satmar just about all Charedi women drive. However, in Israel, most Charedi women don't drive and there is a certain stigma associated with driving. There is an interesting article on Ynet today discussing this חרדית, מי נתן לך רישיון? which is worth reading.
A few days ago in shul I happened to see a new sefer regarding chinuch of girls,האתגר והשליחות לבת מלכות חלק שני (printed this year in ירושלים). There was a whole section about women and driving and how it is assur. In short, driving is not צנוע. There was a whole list of "issues", including that when stopped at a redlight the driver needs to make eye contact with other drivers and pedestrians which of course in not צנוע. In addition, to drive you need to take lessons, get gas, service the car, call a tow truck if stuck, etc. The people who work in these occupations are generally "low lifes" according to the author of the sefer and not the kind of people a בת מלכות should associate with.