Try Malaysia Food Street at Resorts World Sentosa for some choice "local" delicacies (if by local, we also take in neighboring Malaysia's street foods, which are pretty similar and share the same roots as Singapore ones):
I'd second M_Gomez's suggestion of Food Republic at Vivocity - I lunched there almost every day when I worked in the vicinity from 2007-2011. Diners are mostly local office workers, so do avoid peak meal-times, especially lunch (12 noon - 2.30pm).
Some of the more popular stalls include Thye Hong for their fried Hokkien noodles, the Indian curry stall which also prepares prata pancakes to go with their large selection of curries, the Nasi Lemak stall, and the Hakka Lei Cha (Thunder Tea Rice ) stall.
There is also a restaurant, Pu Tien, within the food court - you should try their Hingwa fried bee hoon (rice noodles) which are to-die for!!
A "Malaysian Food Street" just opened at Resorts World Sentosa. There is a lot of overlap between the street foods of Malaysia and Singapore, so you will find things like claypot chicken rice (a very different beast from Hainanese chicken rice), yong tau fu (beancurd and veg stuffed with fish paste), Penang char kway teow, wanton mee, etc. They have apparently persuaded a few famous Malaysian hawkers to set up shop there, so it will be worth checking out how well they can replicate the food in new surroundings.