Best cities to live in Iraq

None, considering the security situation.
I would strongly advise to live in agated compound with high security provided by an internationally recognized company.
If your company does not provide proper secured accommodation, best option would be to live in another company's compound (eg Halliburton camp near Basra / Zubair field).
That's my French opinion after 3 years in Iraq ;-)

If you're asking the question "Where in Iraq should I live?" then there is only one possible answer for you, Erbil, the capitol of the Kurdistan Regional Government (KRG) in the north. It's pretty safe and now days fairly cheap. But if you're looking for work, now is probably not the time. Most people who have work haven't been paid for months. Other cities that are safe within the KRG are Dohuk and Sulaimania. While they're great to visit, there isn't much of a reason to live there unless you have friends, family, or a job in one of those cities. If you do, then you probably wouldn't be asking where you should live.

It would be a terrible idea to go to any part of the rest of Iraq unless you have friends and family there, a job with a major company (and as stated already, in a compound), or you really really know what you're doing.

As for living in Erbil, you could find a furnished house for around $800-$1200 per month, depending on the neighborhood (Ainkawa, Bakhtiary, Harsham). Some of the really nice gated communities like English Village, American Village, or Empire usually start around $1500 if you're lucky, and can go on up from there in price.

Im working in Erbil and havent seen other cities. I can say that Erbil is safe place. You can find here what ever you need but not every time cheap and quality. And sometimes can be problem to find clean restaurant taxi or hotel. My main idea about Erbil is good for to work but not to live.

Hi There ,
My self krunal Sonvane .
i am living in Iraq from more than two years and as per my suggestion to live in Iraq the best place is Erbil and Sulaimaniya which city comes in Kurdishtan region of Iraq .  its well developed and more safer city then any other city in Iraq , Because both cities are to live with family is more easy to get all what you want .

I have recently been transferred to Erbil. The city looks ok, at least much better then any city in south. Now I am planning to move my family as well.