Abra
You must take an abra at least once while in Dubai. These are small ferries carrying about 20 people from one side of the Creek to the other. The fare is AED 1.00 (about EUR 0.20). If you ask for the price they may say AED 10.00 and chance their arm but just pay 1.00.
Bus
If you are not in a hurry it can be good to get round Dubai by public bus (which cost just AED 1.50 (about EUR 0.30) for a journey from trade fair site to Bur Dubai bus station or to the Gold Souk bus station on the Deira side. The buses also run along Jumeirah Beach Road and it is posisble to get to all the way to Wild Wadi from the bus station. Men and women are segregated on the buses.
Taxi
By far the most convenient and most popular way to get round Dubai is taxi (which can be very cheap as petrol/gasoline is cheap) . Taxis are metered.
Metro
A mass transport system (mostly overground) is being built and will be opened with a limited service later in 2009. Will they have segregated trains for women and men? This is one of the big questions for this multi-billion dollar project...