Safety is always one of the biggest concerns when women travel alone or in a group, rest assured that Dubai is very safe compared to other parts of the world. This is a city that you could travel to alone. I would recommend booking your hotel or furnished apartment before arriving as this will relieve any hassle when you get here.
Although this is a Muslim country, the rules regarding dress are not as strict as other countries. Keep in mind that the majority of local women will be covered up in an Abaya (the long black "coat", usually made of a fine material)and Shalla (the black or coloured headscarf). I would recommend light trousers, jeans and skirts for shopping, sight-seeing and wandering around. Shorts, swimsuits and bikinis are fine on the beach and in and around the hotel. You can wear shorts anywhere but you WILL get some stares (a lot probably). Personally, I find it uncomfortable to wear shorts around town but will wear a short skirt to a bar or restaurant at night. I guess you should judge for yourself and these are just guidelines.
Taxis are the main way to get around unless you rent a car. These are safe. There are three main taxi companies and you can flag any taxi down in the street. All taxis are clearly marked with the company name and the number of the vehicle should you run into any difficulties.
Shop. You can shop for just about anything here. It is a definite must in Dubai. Gold, electronics, fabrics, furniture, pearls, carpets, shoes and lots of clothes. You can find just about anything here even fur coats - a definite no in the summer!
There are literally hundreds of bars, restaurants and nightclubs. No matter what your taste in music, food or entertainment, you can find it here. You could take a least one culinary trip around the world in Dubai. I do have my favourite places to go, I'll put those in the next entry!