Scales to Play Over Caug
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 Caug chord contains the notes C – E – G#. Any scale that includes every one of those 3 tones fits over it — from the root C, that means C Whole Tone, C Spanish and C Persian. They differ only in the notes between the chord tones, and that is where each scale gets its color.
Caug is not diatonic to any major key — it is a chromatic color chord, so treat it as a moment of tension: lean on its chord tones from the scales above, then resolve to the surrounding key when the progression moves on.