1) The Shulchan Aruch (268:9) rules that when Shabbat coincides with Yom Tov, we do not make mention of Yom Tov in the "Beracha Me'ein Sheva" ("Magen Avot…") recited on Friday night. When this Beracha was instituted, it was intended to be recited specifically on Friday night, and not on the eve of Yom Tov. As such, this Beracha does not relate at all to Yom Tov, and therefore we need not make any reference to Yom Tov when we recite this Beracha; we conclude the Beracha simply with "Mekadesh Ha'Shabbat," rather than "Mekadesh Ha'Shabbat Ve'Yisrael Ve'ha'zmanim."
2) Similarly, the Shulchan Aruch later rules (284:2) that when Shabbat coincides with Rosh Chodesh, no mention of Rosh Chodesh is made in the Berachot recited after the reading of the Haftara. Since the occasion of Rosh Chodesh does not require the reading of a Haftara, and the Haftara is read strictly because of the occasion of Shabbat, we need not mention Rosh Chodesh in the Berachot of the Haftara. Although the Rambam (Rabbi Moshe Ben Maimon, North Africa, 1135-1204) rules that in such a case we must, indeed, mention Rosh Chodesh in these Berachot, common practice does not follow this position of the Rambam. This is as opposed to a case of Yom Tov that coincides with Shabbat, when we mention both Shabbat and Yom Tov in the Berachot recited after the Haftara.
3) Regarding a case when Yom Kippur coincides with Shabbat, the Shulchan Aruch rules (623:3) that one mentions Shabbat even in the Ne'ila service, despite the fact that this prayer is required due to the occasion of Yom Kippur, and not because of Shabbat. Since the day obligates one to recite four prayers, one mentions both Shabbat and Yom Kippur in all four prayers recited.
