I do not think that most fellowships are not all that competitive; however, I would follow that with the statement that fellowships at well known institutions are competitive.
Here is the scoop:
Clinical Neurophysiology: Not entirely competitive, a number of neurologist will do this fellowship
Epilepsy, sleep, neuromuscular, etc.: Pretty much the same training you'd get as a clinical neurophysiology fellow, just more focused toward one particular area of electrodiagnostics.
Neuro-opthalmology: Wide open! Completely non-surgical field of ophthalmology, hence, opthalmologist don't want to do it. Programs are begging neurologist to take these spots.
Stroke, Vascular, call it what you want: Becoming competitive. Many people are interested doing a fellowship in this specialty so that they can move on and perhaps do further fellowship in neuro-intervention hoping to cash out.
Neuro-ICU: Competitive and a ball buster of a fellowship! Pretty much running an ICU: vent management, lines, codes, etc. Its just that this particular ICU is full of neuro and neurosurgery cases. Again, some programs tied into a stroke fellowship and people do it hoping to move onto neurointervention.
Neuroimmunology, Neurooncology: Do you really need to do a fellowship to see all MS patients? No. But if you do a fellowship in any of these two areas, the difference between you and a general neurologist is that you'd be highly engaged in research in these areas. Not competitive.
Movement Disorders: Up and coming. Again, like neuroimmunology and neurooncology, more focused patient population and research. However, the big thing now is EMG guided injections to treat dystonias.
Headache Medicine: not competitive right now at all. Some recent increase in interest from some as some pain management techniques are being used in this branch.
Pain management: traditionally done my anesthesia or PM&R. Not excluded from this if you are a neurologist, but perhaps at a bit of a disadvantage when applying. Now becoming quite competitive. Many people are starting to see how much money pain specialist are raking in.
Behaviorial neurology, Cognitive neurology, Neuropsychiatry, call it what you want, all the same. Not competitive. Lots of fascinating patients to see and really neat stuff to learn about. But again, a research oriented fellowship.
Neuro-AIDS: Again, research oriented. Was a big deal back when HIV was hitting the brain hard. Now that those ID guys have better antivirals, neuro complications of AIDS are dwindling significantly.