The only true sub-specialty of neurosurgery (one with its own board certification) is pediatric neurosurgery.
After completing a 7-year residency a neurosurgeon should be able to perform all neurosurgical procedures. That being said, many neurosurgeons, especially those wishing to choose the academic route, will do a 1 to 2 year fellowship to gain more proficiency, but it is not required. Available fellowships include vascular, tumor/oncology, skull base, pain, functional, peripheral, and pediatrics. The other aspect that determines whether a neurosurgeon will complete a fellowship is the exposure during their residency. Peripheral nerve surgery is an example of a very, very small field within neurosurgery and therefore most graduate of a neurosurgery program would not have enough exposure during their residency to be competent at these cases and would likely need a fellowship.
Currently, the most lucrative subspecialty of neurosurgery is spine, especially in private practice. Spine fellowships are typically one year. But again, unlike orthopedic surgeons, a spine fellowship is not required for a neurosugeon to focus almost exclusively on spine work. Academic spine is a different matter and a person would likely need to be fellowship trained before being considered for a position at an academic center as a spine surgeon.