Every fingering that fits a hand
x332
easy
877x
easy
8776
hard
877a
hard
8bba
hard
What is true of it
D♯aug is D♯, F𝄪 and A𝄪 — 1, 3 and #5 counted from D♯.
F𝄪 and A𝄪 are double sharps, and they have to be: every letter from D♯ upwards is used once, so the 3 cannot be written any other way.
The easiest of the 5 fingerings below is x332: 3 fingers, held at the 2nd fret.
None of them uses an open string, so this chord moves up the neck unchanged — the same shape 1 fret higher is the chord a semitone above.
1 string stays silent.
Where its notes are on the neck
The brass dots are D♯, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
D♯aug/F𝄪
0332
D♯aug/A𝄪
433x
An inversion is the same chord with a different note lowest. The name changes to a slash name because the bass is what a listener hears as the foundation.