As a woman I feel pretty safe here, even walking at night but then again I still use common sense and pick the areas I go to carefully. I wouldn't walk around in Bur Dubai or Deira alone at night, just like I wouldn't walk around alone at night in Oz...
Yes women can have a pretty good, safe life here- but not for a minute am I under the illusion that sh!t doesn't happen here, just because it's not in the papers, doesn't mean it didn't happen. Once you've been here awhile and talk to some of the people around (shop girls, staff in bars etc) you hear some stuff that makes you a bit more cautious than perhaps Sassy's post would suggest you be.
Sure if you're always around your husband, have someone to drive you everywhere and live in a nice gated compound I'd bet you're safer here than anywhere else. Reality is, people can come and go in most communities as they wish without any checks (have seen it myself) let's not forget awhile back those people hanging around kids in a certain housing development and a principal of a school admitting several students had been approached by strangers. Some people seem to think Dubai is a magical fantasy land where nothing bad happens, use the same common sense as you would at home (if not more, because you don't know the place well when you first come here) don't get sucked in by the idea that it's 100% crime free- because it's not. The police are on the job but you only have to pick up 7 Days to read about the latest killing or drug bust to know that crime does happen here. Maybe not violent petty crime like bag snatching but sneakier stuff, so yes as a woman I would say enjoy but be careful. Not all women live with family here either, yes you can have a lot of freedom as a single girl too but then there are added dangers.
Talk to the wrong person, wear that dress just a bit too short, have one too many drinks.... and there's plenty that can happen to you. I've seen too many girls have close calls on boozy nights out to want to bother with it myself, I prefer to enjoy a beverage or two at home or at dinner with friends, being a woman and being intoxicated in Dubai (aside from being illegal) is just asking for trouble. By all means have your fun but don't believe that stuff doesn't happen just because it's the Middle East.
Try googling women's refuge/rape crisis center in UAE and you'll see what I mean.
If you read into a lot of the news stories these horror stories seem to mainly stem from the kind of people the victims were involved with, so I guess what I'm trying to say is enjoy & live your life but be aware of the pitfalls and be careful who you hang around with/where you go. Be aware of the local laws (and where they might work against you) and keep a low profile.
I can't really say that Sassy's picture of having people available to cater to such things as washing my car or carry my groceries really appeals to me, if you're working with a family then sure but as long as I'm able I'll do my own shopping thanks. Oh yes and I caught a bus last week. Not the most awesome experience but not the horror some people make it out to be. Sometimes you just got to put away any snobby attitude about the people around you and remember that they are just people trying to get by like you are. And unlike you, maybe can't afford a taxi or even the metro so the bus it is.