Looking for a drama or movie (preferably slice-of-life) in which a main female character has had a not-so-cute past -- or present, e.g. gangster; runaway; problematic teen; prisoner; refugee; soldier; bawd; street kid; widow; mistress. Any suggestions?

Nagareboshi
W no Higeki

If you don't mind old jdramas:
Lipstick
Kono yo no Hate
I think that Cinderella's Sister, the role of Moon Geun Young is perfect !
Heart to heart but I don't know if she can be fitted in one of those categories
Doctors she was a problematic teen 
One fine day (I watched it just for the casting)
Bad love
Secret (kdrama)
Gokusen
My lucky star
Doctors (K-drama)
Monster (K-movie)
Kenja no Ai
Deep Love

does victimized or traumatized characters count?
Mars (I prefer the Taiwanese version over the Japanese one)
Angry Mom 
Secret Message