The first one is my favorite, by far -- a song they sang a few times on the show, and included it, as well, I believe, when they did a reunion show. I tried to find a clip of that because it included Ron Howard singing along, but alas couldn't. It's the fight song for the local high school, Mayberry Union High. I always found it very wistful, most especially in how it's used in an episode where they used it for the first time, coming at the very end of a bittersweet high school reunion. Things hadn't gone the way Andy had hoped, as a former girlfriend came back to town, though they resolve things in a tender way. As the two men clean up afterwards, Andy starts to sing and Barney joins in. They brought the song back for this later episode below, which I believe was a class reunion story, as well. If memory serves, Barney comes back to town (Don Knotts had left the show by then), thinking that the news his former long-time girlfriend Thelma Lou wants to tell him is that she wants to reignite their relationship, only to find that it's to tell him she has gotten married. And at the end of the show, a reflective Andy and Barney find themselves alone on the front porch.
This is just one of those kinds of little songs that I dearly love -- helped further because I also happen to love harmony.
I'm going to include the "proper" TV variety show version, sung by a character named Keevy Hazelton, who I believe was a local kid from Mayberry. (It's when he goes back home that he hears Aunt Bee and Clara's song.) The reason I'm posting this, as well, is because the actor who plays Keevy here is Jesse Pearson -- who three years earlier had played Conrad Birdie in the film version of Bye Bye Birdie.