Every fingering that fits a hand
x656
easy · no 5
b9b9
hard · no 5
b9bx
easy · no 5
b9bc
hard · no 5
What is true of it
F♯m6 is F♯, A, C♯ and D♯ — 1, b3, 5 and 6 counted from F♯.
These are also the notes of D♯m7♭5/F♯ and E♭m7♭5/F♯. Which name is right depends on what is underneath and what came before — the notes alone do not decide.
On an instrument that cannot reach all 4, the 5 goes first, in that order.
The easiest of the 4 fingerings below is x656: 3 fingers, held at the 5th 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.
It leaves out the 5. 4 strings cannot carry 4 notes, and the 5 is the one this chord can lose without changing what it is.
1 string stays silent.
Where its notes are on the neck
The brass dots are F♯, the root. Everything else is a chord tone with its degree printed in it.
With a different note underneath
F♯m6/A
2320
F♯m6/C♯
6656
F♯m6/D♯
x320
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.