Update. I am convinced it is a walking stick, not a drum or dance. Here's why: In the song the lyrics are "Singing skin a ma rink a doodle on me auld rigadoo." there are variations on this of course, but I think what the meaning could be is that he's tripping on his walking stick. It's getting in the way sometimes, therefore he's "Singing skinnin' me wrinkled doodle on me auld rigadoo." Obviously, the doodle is the part of his anatomy we shant discuss.