Answer: The Gemara (Taanis 29a) writes that the Beis Hamikdash continued burning throughout the tenth of Av. Therefore, the Shulchan Aruch (OC 558:1) writes that one mustn’t eat meat or drink wine on the tenth either. The Rema, however, limits this until chatzos, noon, of the tenth of Av. The Mishna Berura (558:2) writes that this applies to the other restrictions of the nine days, too, including laundry, haircuts and bathing.
The Rema (OC 558:1) writes that when Tisha B’av is observed on the tenth of Av due to Shabbos, however, one should still refrain from meat and wine until Monday morning. The Mishna Berura (558:4) notes that the other restrictions such as haircuts are permitted that night, though. Thus, the Shaarei Teshuva (558:4) writes that those who are accustomed to avoid haircuts that night are mistaken.
While the Mishna Berura writes that one would normally have to wait until chatzos of the following day to hear music, he writes (Shaar Hatziyun 558:4) that one may be lenient when Tisha B’av is observed on the tenth and one may listen to music that night at a pre-wedding dinner (See Nitei Gavriel, Bein Hametzarim 96:15).
R’ Yehoshua Neuwirth (Shemiras Shabbos Kehilchasa 62:49) adds that when Tisha B’av is postponed, one may say shehecheyanu on new fruits on Sunday night. While he notes that R’ Yechiel Michel Tukachinsky (Luach Eretz Yisrael) forbids one to have a haircut or shave that night, he disagrees, writing that there is no source for this.
In conclusion, when Tisha B’av is postponed due to Shabbos, one may shower, shave, wash clothes and listen to music, etc. on Sunday night, though one should not eat meat or drink wine unless it is at a seudas mitzva.