Scales to Play Over F6
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 F6 chord contains the notes F – A – C – D. Any scale that includes every one of those 4 tones fits over it — from the root F, that means F Major, F Major Pentatonic, F Rock 'n' Roll, F Lydian, F Mixolydian, F Dorian Bebop, F Mixolydian Bebop, F Half Whole Diminished and F Gypsy Major. They differ only in the notes between the chord tones, and that is where each scale gets its color.
F6 is also diatonic to 3 major keys: Bb major (relative minor Gm), C major (relative minor Am), F major (relative minor Dm). 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.