Lychees! Ok so maybe not exactly a "SEA" fruit, but definitely an asian one. Generally air flown from China. Generally brown-ness on the outside means it hasn't been stored at the right temperature and might be "se-kor" (spoiled) on the inside. Fruit should not have any dents visible from the outside. Try to eat one before you buy.
Jumbu-- "Rose Apple" (tastes nothing like roses but is like a very airy apple)
Where to buy these things
The market works well, but if you're really into fruit (and have a friend with a car in Sg) then you should go to the Pasir Panjang Wholesale Center.