Which of these is grammatically correct in response to "What is your nationality?"; “Singapore” or “Singaporean”?
According to the wikipedia page for Singapore, the proper demonym in this case is "Singaporean".
Admittedly, you see plenty of patterns emerge for demonyms but there is no guaranteed rule, which means you simply must check if you're unsure. Just know that the proper term for what you're looking for is "demonym."
Both "Singapore" and "Singaporean" are incomplete sentences, that is, they lack a verb. (Incomplete sentences are widely used in English; using them is not an error.) To make them complete, we would add a subject and a verb to know for sure what's coming on there:
I am from Singapore.
I am Singaporean.
Both are perfectly fine responses, and in this case, you are just making them incomplete by throwing away unnecessary information.