Only in Saudi Arabia women should cover up but as far as I know its not compulsory to cover up in any other country. However a women traveler to the Middle East should wear modestly as a kind of respect to the norms prevailing in the country she is visiting.
http://www.egyptholidaysdirectory.com about 7 years ago
Only Saudi expects you to cover.
In the UAE, you would be expected to dress modestly and appropriately for your surroundings - so no bikini tops and sarongs for the mall, but these are perfectly acceptable on the beach or at the pool.
It is very westernised here, but this shouldnt lull you into thinking that anything goes as you will get stared at and you will make others very uncomfortable.
In the evening, to be honest, everywhere has so much AC that you have to cover up! Jeans, t-shirts, long sleeves just to brave the chilly AC!! For work, I wear a suit, as I would in any business situation.
To enter a mosque (for sightseeing) then you will have to ensure that you cover your hair and that your arms and legs are not bare, but most (big) places will have something for you to wear. about 7 years ago