Generally, Singapore and Malaysia have slight variations of the same list of foods - but, personally, I go for these hawker food items in Singapore:
Curry noodles from either Heng Kee or Ah Heng (both at Hong Lim Food Centre).
Hainanese chicken rice from Boon Tong Kee, Balestier Road. Hainanese chicken rice in Singapore are generally much better than those in Malaysia.
Rojak from Whampoa Food Centre, also at Balestier Road.
Chwee kway (steamed rice cakes) at Jian Bo Shui Kuo stall, Tiong Bharu Food Centre. The food centre also has good stalls selling Hainanese chicken rice, roast pork, etc. Also, Koh Brothers Ter Huang Kiam Chye has the rare Teochew pig’s innards/salted vegies soup which Singaporeans drive across the island for.
Chili crabs (you can’t find the same taste in KL or anywhere in Malaysia, nor the quality). Get them at Long Beach on Dempsey Hill, Palm Beach at One Fullerton or No Signboard at the Esplanade.
Nyonya food. The version in Singapore is totally different from the ones in KL (which is Penang-influenced) and, although similar to Malaccan ones, is much superior (quality-wise) to what you’d find in Malacca. Good Nyonya eateries include True Blue on Armenian St, PeraMakan at Keppel Country Club, Guan Hoe Soon on Joo Chiat Place and Ivins at Binjai Park, Bukit Timah.
These are some food items which you can try for a start.