Scales to Play Over A#sus4
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#sus4 chord contains the notes A# – D# – E#. Any scale that includes every one of those 3 tones fits over it — from the root A#, that means A# Major, A# Natural Minor, A# Melodic Minor, A# Harmonic Minor, A# Minor Pentatonic, A# Blues, A# Dorian, A# Phrygian, A# Mixolydian, A# Dorian Bebop, A# Mixolydian Bebop and A# Spanish. They differ only in the notes between the chord tones, and that is where each scale gets its color.
A#sus4 is also diatonic to 5 major keys: Bb major (relative minor Gm), Db major (relative minor Bbm), Eb major (relative minor Cm), F# major (relative minor D#m), Ab major (relative minor Fm). 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.