I'm trying to find a video which I believe is from the 80's, female black singer walking along a street with lots of people milling about, it was shot in black and white and she is wearing a hat like a beret and a long coat?

Could it be Massive Attack - Unfinished Sympathy? http://www.youtube.com/watch?v=ZWmrfgj0MZI

How about Anita Baker?

I know this doesn't fit all the criteria, but the video that came into my mind was Alesha Dixon 'Breathe Slow'. It's in black and white and she does eventually walk through the street wearing a hat of sorts :)