Dubai is just across the Gulf from Iran, so the supply of Persian carpets is considered one of the best outside of Iran. Prices are also good if you know a bit about carpets and bargaining. If you don't, this is a good opportunity to take the time to learn about designs, quality, origins, and of course prices.
The Blue Souk (Central Market) in Sharjah (below) is famous for its carpet and gold shops. Be sure to visit the Tabriz Carpet Exhibit and talk with Attif Hellal (Right). Mention Dubai Top 10.com and you'll get a good show and low prices.