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.
A A#6 chord contains the notes A# – D – F – G. 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. Start with the first scale listed: it is the most common choice, and the others color the same chord tones differently.
A#6 is also diatonic to 3 major keys: A# major (relative minor Gm), D# 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.