It's always singular with no article (idiomatically, you can only be spoilt for choice, not spoilt for the choice or spoilt for choices, etc.). The meaning is always unable to choose because there are so many possible good choices (that definition is Cambridge Dictionaries, but they'll all be similar).
In practice, the idiom is usually used with positive connotations, since we tend to think of having more choice / greater freedom to choose as a good thing. But as the New York Times reported several years ago...
There is a famous jam study [which concluded that] in reality, people might find more and more choice to actually be debilitating.
My guess would be that the origin of the expression was (perhaps jokingly) "literal" - your ability to make the best choice may be adversely affected (spoilt) by having too many options to consider. But because of the familiar "meaning reversal" as used today, if you did use the idiom when talking about the effect noted in that jam study, it would effectively be facetious. So it wouldn't be surprising to see it in "scare quotes" in such contexts, alerting the reader not to understand the expression as it's normally used.
With acknowledgement to @Andrew's comment, I should just point out that (however it's spelt - spoilt or spoiled), the expression is many times more common in British English than American. But I expect most reasonably articulate Americans would be familiar with it though, even if they might not be so likely to actually use it.