Scales to Play Over A#6
Every chord is a subset of several scales: any scale that contains all of a chord's tones fits over it. Pick a chord to see the matching scales from its own root, the major keys the chord belongs to, and full fretboard diagrams for every scale on guitar.
An A#6 chord contains the notes A# – C## – E# – F##. Any scale that includes every one of those 4 tones fits over it — from the root A#, that means A# Major, A# Major Pentatonic, A# Rock 'n' Roll, A# Lydian, A# Mixolydian, A# Dorian Bebop, A# Mixolydian Bebop, A# Half Whole Diminished and A# Gypsy Major. They differ only in the notes between the chord tones, and that is where each scale gets its color.
A#6 is also diatonic to 3 major keys: Bb major (relative minor Gm), Eb major (relative minor Cm), F major (relative minor Dm). In a song from one of those keys you can also solo with the key's own major scale or its relative minor pentatonic across the whole progression, not just this chord.