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