The best stores for shopping new or used camera gear are almost all located in the Peninsula Shopping Centre, Peninsula Plaza & Funan Centre. All within 5 minutes walk away from one another so it's a one-stop place for your camera shopping needs. Funan Centre is also a good place for buying laptops.

Nearest MRT: City Hall (about 5 mins walk away)
This area is also walking distance away from Boat Quay, Esplanade etc, which are scenic tourist spots for you to test your new gear.

Here's a list of recommendations for shops to check out.

1. Cathay Photo Store
111 North Bridge Road #01-11-14 Peninsula Plaza. Phone: +65.6337 4274
Reliable and good service. No pushy sales guys.
Reasonable prices. With GST refund, prices are comparable to B&H online prices (if you include shipping costs)

2. The Camera Workshop
3 Coleman Street, #01-31, Tel: 63361956 / 6336 7808
Great place for finding used camera gear. While the prices of their used gear are not cheaper than Ebay, the good thing about buying used items here is that you get to test the item before use. The shopkeepers are friendly and not pushy.
They also carry parallel imports for brand new cameras (slightly cheaper but only offers international warranties, no local warranties)

3. John3:16
#05-46, Funan Digitalife Mall, 109 North Bridge Road. phone +65 63372877
A small christian family-run shop with very honest to goodness business ethics. As this is a small shop, they don't keep large quantities of stock. However they can easily order what you want and call you when the stock arrives.
Take a virtual tour of their shop

4. Prme Camera Centre
111, North Bridge Road, #01-15 Peninsula Plaza. Phone +65-63347581
Mostly used gear. It's the next best place to look for used cameras after The Camera Workshop.
Conveniently located right next to Cathay Photo, it's worth the time popping in to take a quick look.

5. Ruby Photo Store
3 Coleman Street, #01-01 / 01-02, Peninsula Hotel Shopping Complex. Ph : 6338 0236 , 6338 4104
A good place for buying studio stuff. Official dealer for Bowens studio lights in Singapore. Also carries darkroom supplies.
They don't sell digital cameras. They carry lomo and diana cameras.

6. Camera Rental Centre
They don't sell cameras or lenses here but they have a huge range of gear for rental at really affordable prices. Good to rent a gear you intend to buy and testing it out for a day before you commit to purchase it elsewhere.
It's about 10 minutes walk away from the shopping places mentioned above.
A straight walk down South Bridge road (the one-way traffic) past Funan Centre. Located at the 2nd floor of an old pre-war shophouse, it's best to call for directions.