Q:

Is it permitted to use beard oil on Shabbat?

Is it permitted to apply beard oil to a beard on Shabbat? I'm assuming not but don't have a source.
For those unfamiliar with beard oil, it is a moisturising oil applied to the beard and skin underneath to moisturise the beard. This stops it becoming dry and brittle, particularly in cold and windy weather. See this Wikipedia arrive for more: https://en.m.wikipedia.org/wiki/Beard_oil

A:

According to Shemirat Shabbat K'Hilchato as cited here, there is a leniency regarding oil, which is how Wikipedia describes this product:

Regarding applying hair cream on Shabbat, Shemirat Shabbat Kehilkhatah
  writes that one may be lenient and apply hair oil, but only in small
  quantities so that one should not come to squeeze it out of the hair.
  However, he prohibits the use of hair cream in any quantity. His view
  is shared by responsa Be’er Moshe. Ma’amar Mordekhai, writes that in
  his day the minhag was to rub the hair on Shabbat with a mixture of
  liquid fats and perfume and states: "I worry that people who do this
  are guilty of breaking Shabbat, as it appears to be prohibited because
  of smoothing (on the cream)." Rabbi Ovadia Yoseph prohibits this and
  writes: "Since the person intends for this lotion to remain on his
  hair, there is no Torah prohibition; nevertheless, it is forbidden
  from the Rabbis."

Please use this info as material to show your Rav and ask him. The citations implies various opinions.
There also may be a separate concern of possible prohibition of gozez which may occur if you accidentally pull out your hair during the process of rubbing the oil in your beard. I have not addressed this issue, here. You can get more info on hair handling and gozez from this article.

