What is the most common type of vascular rings in children?

The basic back bones of the development of the vascular system are the dorsal aortae and ventral roots, and 6 aortic arches that join the two, on the right and left.

Regression of some of the arches in central to the normal development of the vascular system.
Only the 3rd, 4th, and part of the 6th arches should persist to contribute to the carotid arteries, right subclavian/aortic arch, and the pulmonary arteries, respectively (figure).

The most common vascular ring (1% of population) results from a right subclavian artery that takes off from the arch of the aorta on the left, due to abnormal regression of the right 4th arch. The subclavian artery passes behind the esophagus.  This type of incomplete vascular ring rarely causes symptoms in children, and surgical therapy, if needed, consists of division of the vessel through a left thoracotomy.

Reference:
Pediatric Surgery.  Coran A. 7th edition