Scales to Play Over A#m
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#m chord contains the notes A# – C# – F. Any scale that includes every one of those 3 tones fits over it — from the root A#, that means A# Natural Minor, A# Melodic Minor, A# Harmonic Minor, A# Minor Pentatonic, A# Blues, A# Rock 'n' Roll, A# Dorian, A# Phrygian, A# Dorian Bebop, A# Half Whole Diminished, A# Gypsy Major and A# Gypsy Minor. Start with the first scale listed: it is the most common choice, and the others color the same chord tones differently.
A#m is also diatonic to 3 major keys: C# major (relative minor A#m), F# major (relative minor D#m), G# 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.