I used to live in UAE a few years ago. It is a nice place for men... As a woman, I recommend you be very careful. Dubai, Manama and Doha are very safe, in fact safer than walking in the middle of IA. Abu Dhabi is not as woman friendly as you expect. It is nice to visit but there are many restrictions as to what a woman can safely do with out rising suspicions from the locals. Obviously you are aware that you must be prudent and prudish regarding dress.
I would spend most of my time in Dubai, there is more to do that anywhere else in the area. I really loved the dune racing and the very fine dining. During the time I was there I never ever worried about anything at all. Never felt unsafe never felt nervous or similar. As a sigle woman I would certainly not walk down the creek area after 10, unless there are many people on the street.
Dubai has a great cab system, they are cheap and everyone speaks English. I recommend you try Iranian food and really enjoy this.
Things to look out for in Dubai.
As an American you are restricted from conducting or doing business with certain countries and companies. Dubai is a world financial center and as such there are many international banks with ATM's everywhere. Make sure, and I mean make sure, you do not use an ATM from an Iranian bank, you will be violating the OFAC (office of foreign assets control) guidelines. I have a friend who upon returning to the US had a lot of trouble, for a while until he could prove he did not intentionally do so.
Be mindful of your smiles, certain cultures that populate this area (with European / westerners) you dont have to worry about but other races and cultures will take this as an invitation to engage you and potentially ruin your evening. A small smile is always ok but a large American style smile is not your best option.
In the bars (always at hotels) you will encounter people who offer you a bag of "herbs" not MJ but some other Persian or North African "tea" as they call it. These are not laced or anything, they will just really make you sick and depending on your body may ruin a few vacation days for you.
DO NOT walk around drunk.... there is Zero tolerance for women who seem intoxicated on the streets. Use a Cab, to move from place to place after consuming alcohol. You can get wasted just like any where else but be aware that this cannot be done outside specially designated areas.
Last but not least... Alcohol is very pricy in Dubai, so be sure upon arrival to stock up on any and all the alcohol you want to enjoy in your room from the arrival duty free shop just before you exit the airport security area. It is cheap and well, you will appreciate being able to have a drink in your room after hours or before you go to one of the many awesome bars or clubs.... Remember to not arrive drunk, just have a few before, in the event you are wanting to party like a rock star at a club.
Have fun... it is one of those trips you will never forget. I love that city!