Hi Crisscross (do you wear your clothes back to front?)
Shopping in the major malls can produce some bargains, particularly at sale time, and the selections can be very wide, however prices are pretty much on a par with Europe. if you are coming here with a strong currency, then things will appear marginally cheaper, but don't expect super cheap deals.
To get the best prices on genuine brands, try Dubai Outlet City (a 15 minute cab ride out of town on the Al Ain Road) which is mostly clothes brands.
For best deals on electrical goods, try Sharaf DG at Times Square Mall on the Sheikh Zayed Road or Jumbo Electronics who have their own big store on Makhool Road opposite the Al Ain Centre. Al Ain Centre itself is good for browsing for cheap computer equipment and gadgets (Its also known locally as Computer Mall) and is at the junction of Mankhool Road and Bank Street.
If you are looking for cheap and cheerful, there is Dragon Mart (another 15 minute cab ride out of town on the Hatta Road) which is a huge collection of Chinese origin products. They are cheap as chips, but will probably break down or disintigrate before you get them back to your hotel!
For the best clothes bargains in town and a real adventure, you need to go to Karama! This is a whole district of small stores selling knock off and fake designer brands. Be prepared to scour all the shops, fight off the hawkers trying to sell you copy watches, copy handbags and copy DVD's...and BARGAIN! You never pay the price they quote or is printed on any label. They expect you to bargain and you should aim to pay 50% - 60% of the asking price. All the goods are dodgy, some are actually quite good quality, much is totally tasteless, but its great retail therapy and you will probably come away with bags and bags of goodies.
Enjoy!
Knight