With Roman numeral analysis you can put sharps and flats on the Roman numerals to show the root is altered.
But, while a altered dominant root with
♭V makes sense a
♯V does not. Here is why...
Usually such alteration are about borrowed chords or mode mixture. For example, in a major key you can have a
♭VII borrowed from mixolydian. When you add flats in the order of the circle of fifths the sequence of altered tones, starting from
C major, is
B♭ E♭ A♭ D♭ G♭ and the corresponding modes in sequence are mixolydian, dorian, aeolian, phrygian, locrian. When you add sharps the sequence is short, it's just one move
F♯ taking it to lydian.
When you get to the end of either the flats or sharps direction the next step would alter the tonic from
C to either
C♯. Things get weird, but if we keep going up with sharps to finally get to the alteration of a
♯V, which starting from
C major would be a
G♯, we first move through
C# locrian, then
So, alterations like
♭VII, for example, make sense, because you can say they are borrowed from the parallel modes phrygian or mixolydian. From
C major those would be
C phrygian or
C mixolydian. But for
♯V you wouldn't even get the root of the chord until
C# phrygian. Where the tonic has changed! But, even then the chord would be
#vo, a diminished triad. To get a diatonic
#V you have to add seven sharps to get to
All this becomes a whole lot clearer when you use the key signature which should be given with Roman numeral analysis. This...
C: I III7 II7 C#:V C:V7 ...uses a
G# major triad but requires a really clumsy tonic change in the middle of the progression. But, if you spell the root as something diatonic to a
C tonic, it will be enharmonic equivalent
C: I III7 II7 ♭VI V7 ...is so much clearer! One tonic with borrowed chords.
Basically, you shouldn't normally see root alterations on
V. They don't make sense. A
♭V would be a diatonic chord in locrian, but locrian is pretty unusual. Not at all common.
Alteration on dominants that you will commonly see are not alterations to the root, but on the chord's fifth and ninth, ex.
G7♭5. I've seen that combined with Roman numerals like