"Who wants ice-cream?" — Should I say "(not) I" or "(not) me"?

Solution 1:

Generally speaking, in English, accusative (also known as “objective”) pronouns (like me) are the “default” form. That is, unless there is a specific syntactic rule requiring use of a different case, such as nominative (I), genitive (my/mine), or reflexive/intensive (myself), in English you use the accusative case.

In the syntactic context where a pronoun is not serving a role relative to an explicit verb, such as when it is the simple answer to a question, or if one is labeling something, such as a photo, accusative pronouns are standard. “Who wants to come?” “Me.” Nominative pronouns are impossible here—you cannot answer the question “Who wants to come?” with “I”, nor would anyone label a photo “I”.

This holds even if negated: “Who wants ice cream?” “Not me.”

If you want to use the highest register, most formal English, however, you should avoid the question of what case to use with pronouns standing alone, and use a complete sentence: “I do not want ice cream.”

Solution 2:

"I" is used when the speaker is the subject of the sentence:

"I kissed Eve."

"Me" is used when the speaker is the object of the sentence.

"Eve kissed me."

Strict grammatical logic says that the same is true when answering questions.

"Who kissed Eve?"

"I did." or just "I."

However in common usage, going back centuries, people frequently use "Me" as a single word answer where "I" would seem logical. It's so common that it can't be considered wrong:

"Me."

..

"Who did Eve kiss?"

"Me."

By way of illustration; the classic nursery rhyme:

"Who killed Cock Robin?"

"I," said the Sparrow, with my bow and arrow"

Adding "not" changes nothing.

"Who kissed Eve?"

"Not I" - agrees with the formal, and still acceptable grammar

"Not me" - agrees with common usage

Both are fine. "Not I" is perhaps a bit ostentatiously literate

"Who did Eve kiss?"

"Not me." -- never "Not I"

Just as with that old chestnut less versus fewer, it's a one way street. People rarely say fewer when less is "correct" (except sometimes deliberately, to annoy). People rarely say "Not I" when "Not me" is "correct".

Bottom line: if you always say "Not me", you'll never be wrong. But you miss out on the chance to be extra rigorous in your grammar.