E minor
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G Major or e minor. The sharp would be F sharp.
A key signature with one sharp (F#) would be either G major or E minor.
Four key signatures have G sharp and E sharp in them: F sharp Major, D sharp minor, C sharp Major and A sharp minor.
An E natural minor scale is written with a one-sharp key signature, so you just write out the scale in semibreves and add an additional D sharp for the harmonic minor, and two additional sharps to the sixth and seventh degrees when ascending in the melodic minor (C sharp and D sharp). You lower the sharps and revert back to the natural minor when you descend through the scale.
There is only one sharp in G minor: the F# in the harmonic and melodic minors. The key signature for G minor is two flats, so the answer to your question is pretty much zero. Its leading tone, however, is F-sharp, always written with an accidental, so I suppose one could consider that there is one sharp in G minor, although not in the key signature. Please also see the related link about the G minor scale.