The Shabbat prohibition of Gozez refers, in its narrowest sense, to shearing wool, but it applies as well to the removal of any substance from one’s skin with an instrument. Thus, for example, it is forbidden to cut one’s fingernails on Shabbat, and cutting fingernails with scissors constitutes a Torah violation. However, if one has a hangnail – meaning, a fingernail that is mostly detached from the skin – and the nail causes him discomfort or annoyance, he may remove the nail on Shabbat with his fingers or teeth. Since the majority of the nail is already detached from his skin, he may tear the remaining part off the skin.
The prohibition of Gozez also applies to removing hair from one’s body, and thus gives rise to the interesting question of whether a person may remove a bandage from his skin if this will inevitably result in the removal of hair. Would it be forbidden to remove a bandage in this case, given the prohibition against removing hair from one’s skin on Shabbat?
This situation falls under the category of "Pesik Resheh De’lo Niha Leh" – an action that will inevitably result in a Melacha (activity forbidden on Shabbat), but the individual has no interest in this result. When one removes the bandage, he obviously does not want the hair to come off along with the bandage. Halacha permits performing this kind of action if the resulting Melacha is forbidden by force of Rabbinic enactment, as opposed to Torah law. In the case of removing hair along with the bandage, this would not constitute a Torah violation, since no instrument is used. Therefore, this qualifies as a case of "Pesik Resheh De’lo Niha Leh" involving a Rabbinic prohibition. It is therefore permissible to remove a bandage from one’s skin on Shabbat even if this will inevitably result in the removal of hair.
It is strictly forbidden to comb or brush hair on Shabbat, as this results in hair being removed from the head. Some Halachic authorities go so far as to consider attending synagogue with combed hair on Shabbat an act of public Shabbat desecration. It is, however, permissible to comb a wig on Shabbat, since the hair is not in its natural location. Therefore, women who wear wigs may comb their wigs on Shabbat, even though this will result in removing some strands from the wig. If, however, the wig is disheveled and cannot be worn in its current state, then combing it would violate the Shabbat prohibition of "Metaken Mana" – fixing, or rendering an item usable. Even though no prohibition of Gozez would be entailed, combing the wig in such a case would violate the Shabbat prohibition of "Metaken Mana."
Summary: It is forbidden to cut one’s fingernails on Shabbat, though if one experiences discomfort due to a hangnail that is mostly detached from the skin, he may pull it off with his fingers or teeth. One may remove a bandage from his skin on Shabbat, even though this will inevitably cause some hair to be removed from his skin. Combing or brushing one’s hair is forbidden on Shabbat, but one may comb a wig on Shabbat, unless the wig is completely disheveled and cannot be worn in its current state, in which case it may not be combed on Shabbat.
