The prohibition of Gozez (shearing) forbids removing hair from one’s head, beard or other parts of the body on Shabbat. The Shulhan Aruch (Orah Haim 302:13) writes that as a safeguard against violations of this law, the Sages forbade using on Shabbat a metal mirror with a sharp edge. It seems that in Talmudic times, mirrors would have a sharp edge so that when one looks into the mirror and sees hairs that need to be removed, he can cut them using the mirror. The Sages thus feared that one who uses such a mirror on Shabbat may remove a hair, in violation of Shabbat, and they thus enacted a prohibition against using such mirrors on Shabbat. The Shulhan Aruch writes, however, that it is permissible to use a mirror that does not pose such a risk. Therefore, it is certainly permissible to use mirrors nowadays, as they are used only for looking at oneself, and not to cut hairs. This applies both to mirrors that are attached to the wall and to hand-held mirrors.
There is, however, one contemporary application of this Halacha. Hacham Bension Abba Shaul (Israel, 1923-1998), in his work Or Le’sion (vol. 2, 24:10), writes that it is forbidden on Shabbat to look at one’s reflection in a metal knife or scissors. It could happen that while sitting at the Shabbat one might look at his reflection in his silverware to check his hair, for example, and Hacham Bension rules that when it comes to knives, which could potentially be used to remove hair, this is forbidden. Thus, although it is certainly permissible to use mirrors on Shabbat, it is forbidden to look at one’s reflection in metal knives or scissors.
Summary: It is permissible to look at one’s reflection in a mirror on Shabbat, but one should not look at his reflection in a metal knife or in metal scissors.