Singapore is quite a small city. If you would like to look around town and do some shopping/market research, the place to go is Orchard Road. Within that one city block, there are numerous malls where everyone shops, from luxury brands to high street boutiques. I suggest you visit Ion, Takashimaya and Paragon. I can't say you'll find anything to buy in Singapore that would be unique or exceptional.
Outside of Orchard Road, perhaps the place that is most unique is Marina Bay Sands, the casino, restaurant, hotel and shopping complex. It's huge, there are many international restaurants within including Wolfgang Puck, Mario Battali and Daniel Boulud's DB Cafe. Also an extremely popular restaurant for something Asian called Din Tai Fung, originating from Taipei, which offers a variety of dumplings, delicious. Take a walk outside along the boardwalk for great Singapore views and visit the Louis Vuitton floating island boutique. At the top of the Marina Bay Sands is a restaurant / nightclub / bar called Ku de Ta, the crazy shaped boat design, go up there for a drink, you'll see the entire Singapore.
If you prefer a great spa experience, you can book and go to The Capella Hotel & Resort on Sentosa Island which is only a 20 minute ride from the airport.
For a taste of local flavours, you could visit Newton Circus, it is what locals would deem hawker stalls. A lot of street vendors offering local cuisine, you can purchase from any of the numerous stalls, it's cheap and delicious.
Michelle Yang is a world traveller, jet setter, MFVD supporter, and a dear friend.