I don't really think of Singapore as "sunny" though it certainly is warm...glad to have you among us.
Welcome! My parents went to Singapore last year at the end of a cruise - knowing they'd be going, I researched shaving goods and found that the Mustaffa Center in Little India was said to have all the good stuff (blades and creams and aftershaves that I can't buy here in Canada). They went, looked around, found the shaving stuff... and came home empty handed. Everything on the wish list I'd provided them was out of stock and the other stuff they did have wasn't particularly cheap, or I already had it. I only found out about the shop called Haniffa[edit: spelling] afterwards - I heard they have a good selection and is a bit cheaper. In case they go again (they loved the place and the F1 GP is on their bucket list), do you know of any other good shops in Singapore for shaving supplies?
Try Apollo sellapas and haniffa textiles. Both are on Dunlop Street. Also, Mustafa has got 2 sections. Entrance 1 gets you to the main store. Entrance 4 which you access by crossing the road has more shaving stuff. Your parents likely went through entrance 1 like most tourists.