A mourner does not lead prayer services on Shabbat or on Yom Tov, including the intermediate days of Pesach and Succot. However, there is a custom that a mourner be given the Maftir and the Haftarah on those days if there is no other person who should receive it.
On Rosh Chodesh there are different customs. In some communities a mourner does not lead the Services at all. In others the mourner leads the Services until Hallel and then someone else takes over. In other communities (mainly the Yekkes) a mourner leads the Services only at night on Rosh Chodesh. On Chanukah the customs vary a little - in some communities the mourner leads the Services until Hallel, someone else takes over for Hallel itself, and then the mourner then carries on after Hallel. In other communities a mourner does not lead the Services on Chanukah at all. On Purim the custom is that a mourner does not lead any of the Services.