Scales to Play Over D# major
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# major 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# Major, D# Major Pentatonic, D# Rock 'n' Roll, D# Lydian, D# Mixolydian, D# Dorian Bebop, D# Mixolydian Bebop, D# Half Whole Diminished, D# Spanish and D# Gypsy Major. They differ only in the notes between the chord tones, and that is where each scale gets its color.
D# major is also diatonic to 3 major keys: Bb major (relative minor Gm), Eb major (relative minor Cm), 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.