The atlanto-axial joint is supposedly a pivot joint. I assumed it would be considered a ball-and-socket joint, based on a) the physical shape of the dens, and b) the degrees of motion of the cranium relative to the spinal column.
A pivot joint is distinct from a typical ball-and-socket joint because it only allows for a single degree of freedom; yaw - to rotate about it's own axis or diameter, like a spindle. A typical ball-and-socket joint allows at least one additional plane of limited movement.
I can think of (or observe) at least three degrees of freedom of the cranium, relative to the spinal column; pitch & roll. Pitch allows you to direct your gaze from the ceiling, to the floor. Roll allows lateral movement in a similar manner, so you can point your left ear to the ground, and then your right.