What is our higher self? Some religious or philosophical systems hold that there is an immaterial divine component to human existence, like the Soul or the Higher Self.

Is there any kind of scientific evidence that such a supernatural essence of the person exists, and if so, can it be located in a specific part of the brain?
Wikipedia's answer is a better answer than I could offer as to what the higher self is, because that's all you've used to define it, and as Wikipedia says, it's "a term associated with multiple belief systems," and probably differs rather widely across the gamut. Regardless of how odd some of those beliefs might get, it probably shouldn't be synonymous with the prefrontal neocortex or basal ganglia, each of which has multiple functions that couldn't simply be equated to what most people mean when they say "higher self."

In one aspect at least, there may be some overlap between the "higher self" concept and the prefrontal cortex. To whatever extent "higher self" means "self-awareness," the prefrontal cortex is probably involved in that...This article offers an interesting image from magnetic resonance tomography, and these quotes [emphasis added]:


  A specific cortical network consisting of the right dorsolateral prefrontal cortex, the frontopolar regions and the precuneus is...associated with self-reflective functions...
  
  By comparing the activity of the brain during...lucid periods with the activity measured immediately before in a normal dream, the scientists were able to identify the characteristic brain activities of lucid awareness.


So note that this is self-awareness in the context of dreaming. The original article (Dresler et al., 2012) is free, and mentions an interesting method for distinguishing lucid from normal dreaming, BTW. As for consciousness in general, we have some pretty interesting questions with richly informative answers here already, including:


What are current neuronal explanations and models of &#39;consciousness&#39;?
Is there a region of cortex which over a period of development becomes the seat of self?
Research suggesting conscious control over brain region activation?
If someone becomes a split-brain patient, which side will &quot;maintain&quot; the continuity in their consciousness?


And as for the subconscious, Wikipedia again gives a pretty decent answer. Since the term is frequently misattributed to psychoanalysis, and may be most popular in pseudoscientific contexts, I prefer to point you directly to the section on Freud's answer and advocate his general practice of avoiding the term altogether (except for the sake of criticizing it). Freud used "preconscious" and "unconscious" in distinct ways that cognitive psychology has adopted somewhat as well, so those terms are generally much more useful and well-defined scientifically...though that's not saying all that much. However, thanks to @ArtemKaznatcheev, we also have a question here that says a little more: Subconscious vs Unconscious

Reference

Dresler, M., Wehrle, R., Spoormaker, V. I., Koch, S. P., Holsboer, F., Steiger, A., Obrig, H., Sämann, P. G., &amp; Czisch, M. (2012). Neural correlates of dream lucidity obtained from contrasting lucid versus non-lucid REM sleep: A combined EEG/fMRI case study. Sleep, 35(7), 1017-1020. Available online, URL: http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3369221/. Retrieved January 25, 2014.