Another theory, just as speculative:
When the place name ends in a vowel, like America or Australia or Canada or India or Indonesia or Macedonia or Malaysia or Nigeria or Asia or Latin America, it is slightly awkward to say phrases like "America Idol" or "Canada Idol", with two consecutive vowels. The phrases like American Idol and Canadian Idol roll more easily off the tongue. (This is the same phenomenon behind the article a becoming an before vowels in English, and behind linking R and other related sandhi processes.)
Because Singapore ends in a consonant sound, there is no reason to use the longer adjectival form, and the place name can be used directly. In support of this theory, observe Pakistan Idol which is not turned into Pakistani Idol.
(This theory doesn't fully explain why Greek Idol instead of Greece Idol, and why Hrvatski Idol is fine, but it does explain why the adjectival forms are chosen even when they are longer, as in Canadian Idol over Canada Idol.)
There is a question on travel.SE at the moment that I think is similar.
Both questions may be formulated as: Is a demonym a noun or an adjective?
On Wikipedia we can find a list of adjectival and demonymic forms of places.
I would suggest a demonym is a noun, so saying Singaporean Idol (demonym) would be a mistake, while Singapore Idol (adjective) is the right form.
In the travel.SE question, the matter is whether we should say New Zealander citizen (demonym) or New Zealand citizen (adjective).