C major scale on Irish bouzouki
No sharps, no flats. Less common in the tradition than G or D, but the O’Neill collections carry a solid set of C tunes, many of them airs and set dances.
Hear the scale, played on the banjo voice
The notes
Every note above sits in first position on a GDAD-tuned Irish bouzouki: nothing higher than fret 5, and each note has one home. Learn the scale as a shape and your fingers know where every note of every C major tune lives before you have heard it.
The chords that go with it
This is the connection that makes session playing make sense. The backer's chords are built from the same 7 notes as the scale, so when the guitar holds C, anything you play from the C major scale fits. Melody players who seem to improvise freely are doing exactly this: decorating the tune with notes that never leave its own scale. Shapes for every chord here are on the Irish bouzouki chord page.
Tunes in C major
78 of the 2701 tunes on this site are recorded in C. A dozen to start on, in alphabetical order - every one has free GDAD tab, playback and the ABC source.
Questions about C major
C, D, E, F, G, A, B, C. On a GDAD-tuned Irish bouzouki the whole scale sits in first position, between the open strings and fret 5.
C, F, G carry most tunes in C major, with Am, Dm, G7 as passing and turnaround chords. Two or three chords are enough: Irish accompaniment stays out of the tune's way.
In an Irish session the chord follows the tune, not the other way round. If the tune is in C major, every note of the scale above fits over its chords, including C and F and G. That is how melody players improvise variations without reading anything: they stay inside the tune's own scale.
This site's collection has 78 tunes recorded in C out of 2701 in total, each with free GDAD tab. The corpus is O'Neill's 1903 and 1907 collections plus Kerr's Merry Melodies, so the balance reflects the tradition as it was written down.