Nitei Gavriel Aveilus 56-58 discusses the different Minhagim. This is my understanding of what he says.
An Aveil should not Daven for the Amud on Shabbos, Yom Tov, and the high holy days. Exceptions are as follows. If there is no one who can Daven as good as the Aveil the Aveil may Daven during these days. In addition if the Aveil is the regular Chazan on the high holy days he may continue to Daven for the Amud on the high holy days. If he always Davened on Shabbos and Yom Tov for the Amud he may also continue to Daven for Shabbos and Yom Tov for the Amud. 56:6.
Erev Shabbos and Erev Yom Tov for Mincha, most Chasidim have the Minhag that an Aveil does not Daven for the Amud. The Minhag of the Ashkenazim is that the Aveil does Daven for the Amud. 56:10.
Regarding Selichos prior to Rosh HaShana and during the Aseres Yemai Teshuva there are different Minhagim. Some say an Aveil can say the Selichos even on Erev Rosh HaShana and some say an Aveil should not say the Selichos and should only Daven Shacharis from Ashrei / U'vo L'Tziyon. 56:15&16.
Rosh Chodesh there are differing Minhagim. Some say an Aveil can Daven everything besides Hallel and Musaf. Others say an Aveil should not Daven for the Amud on Rosh Chodesh at all, not even for Maariv. Then there are those that say an Aveil should Daven Mincha and Maariv for the Amud, however not Shacharis, Hallel and Musaf. 57:1.
Chol HaMoed an Aveil should not Daven for the Amud. 57:4.
Chanuka an Aveil Davens all Tefilos besides Hallel for the Amud. However there are those that have a Minhag not to Daven at all for the Amud on Chanuka. Then there are thise that Daven Mincha and Maariv for the Amud, however they do not Daven Shacharis for the Amud. 58:1.
Purim most people do not Daven at all for the Amud. Then there those that Daven from Ashrei / U'vo L'Tziyon. 59:1.
Shushan Purim some Daven all the Tefilos and some do not. 59:3.
Purim Katan, some say that the Aveil does not Daven at all for the Amud, however most have the Minhag that the Aveil does Daven for the Amud. 59:4.
The Shaalos U'Teshuvos Megidos 83 (Pri Megadim) and Biur Halacha 132 say that any day we do not say Lamnatzaiach by Shacharis the Aveil does not Daven for the Amud.
Some say an Aveil should not Daven for the Amud the entire Tishrei and entire Nisan unless the Aveil has Yarzheit on that day. 60.
Pesach Sheni, Lag B'omer, Chamisha Asar B'Av, Chamish Asar B'Shevat, from Rosh Chodesh Sivan until Isru Chag, Tisha B'Av - all have different Minhagim where some do all, some do none, and some do some. 60.