Climbing is still a relatively young sport in the UAE, and retailers tend to carry a limited selection of equipment. Your best bet is to call ahead if you're looking for something in particular.

Map

Indoor walls are indicated by pushpins. Outdoor walls are indicated by a Trad, Sport, or DWS icon. Shops where you can purchase climbing gear are indicated by a yellow dollar sign. Click on an icon to get more information.

You might want to open the map in a new window.

Shops

Decathlon

Possibly the best selection and prices in the UAE. Shoes, carabiners, harnesses, draws, slings, belay devices, rope (including standard 60m, half, and by the meter), cordelette, loose chalk, and more. Very neat and organized. The store also has lots of other interesting gear like scuba and riding tack. 
Location: Mirdif City Centre, Dubai.
Hours: Sunday to Wednesday: 10am to 10pm, Thursday to Saturday: 10am to Midnight
Phone: 04 283 9392

Go Sport

A sparse collection and high prices, but one of the few places to get the UAE Rock Climbing book. Has shoes, harnesses, chalk balls, chalk bag, and a small assortment of carabiners, draws, and trad gear.
Locations: Ibn Battuta Mall, Mirdif City Center, Mall of the Emirates (particularly poor selection).
Hours (Ibn Battuta & Mirdif City Center): Monday, Tuesday, Wednesday, Sunday from 10:00 to 22:00, Thursday, Friday, Saturday from 10:00 to 00:00. 
Hours (Mall of the Emirates): Monday, Tuesday, Wednesday, Sunday from 10:00 to 23:00, Thursday, Friday, Saturday from 10:00 to 00:00
Phone (Ibn Battuta): 04 368 5344
Phone (Mirdif City Center): 04 284 3034
Phone (Mall of the Emirates): 04 341 3251

Picnico (CLOSED 2016)

A small store specializing in odd hobbies like climbing, fishing, camping, and more. Carries Red Chili shoes, a very small display case of metal, and two or three different harnesses (Petzl and Wild Country, I think). Really not a whole lot to see. Closed as of 2016.
Location: behind EPPCO Petrol Station near Jumeirah Beach Park.
Hours: Sat-Wed 9am-9pm; Fri 11am-7pm (?)
Phone: 04 283 9392 (?)

Adventure HQ

Opened June 2011 with a climbing wall and high ropes. Has possibly the best selection and a neatly organized selection of gear, from ropes and harnesses to cams and slacklines. Prices tend to be higher. The climbing wall is all about autobelays and jugs right now, but with good routesetting it could become interesting.
Price: 50 AED/30 min
Location: Times Square Dubai.
Phone: 800 ADVENTURE (800 238368873)

Dorell Sports Climbing Wall

Expected to get equipment in around July 2011.
Location: Near World Trade Center, Dubai. Totally unmarked from the front. Find the Block C of the Dubai Trace Center Hotel Apartments. Walk through the unlikely lobby to the back, outside. 
Hours: 1:30p - 10:30p daily (but call them, hours vary)
Phone: 050 784 1789 or 04 306 5061

Global Climbing

An outdoor (and climbing) gear distributor for many of the retail stores in the UAE.
Also check out their clearance list.

Mountain Extreme

Primarily a climbing wall, but also has a gear shop.
Location: Dubai Business Bay
Hours: Every day 10:00a-10:00p
Phone: +9714 446 6080

TRAKS PRO

A Petzl & Beal distributor. Not really set up for retail operations, so call ahead.
Phone: 04 885 9980