The Bur Dubai Souq, across from Deira and right next to the abra station (this is where the boats drop you off when you cross the creek) had some of the best deals we saw in Dubai. Lots of knockoff watches, handbags, as well as beautiful textiles, rugs at rock bottom prices (after haggling). I picked up some nice shirts for $2 each.
The Karama Shopping Center / Market is a little difficult to get to (due to the scarcity of cabs) but has a lot of great shoe shops, more gold shops, and lots of clothing. It had a very local feel to it and was a good experience.