(For my comrade-in-blog Mochassid)
The prophet Yishayahu tells us (58:13): “ve-khibadto – and you shall honor [Shabbos].” The Gemara (Shabbos 113a) applies this oblgiation to the clothes one wears: “Your clothing for Shabbos should not be like your clothing for weekdays.” Note that the Gemara does not say that you should change your clothes for Shabbos but that your clothes for Shabbos should be different from those for weekdays. The Ben Yehoyada (ad loc.) explains that one might have thought that it is sufficient to have different clothes for Shabbos. Therefore, the Gemara teaches us that this is not enough but rather one must wear clothes that are sufficiently special that onlookers will recognize that the garments are for Shabbos.
This halakhah is recorded in Shulhan Arukh (Orah Hayim 262:2-3). The Magen Avraham (2) writes in the name of the Arizal that it is best not to wear any weekday garments on Shabbos, and this is brought down in the Mishnah Berurah (5) and the Arukh Ha-Shulhan (3). The question arises regarding a mourner during shivah: Should he wear Shabbos clothes or not? The Rema writes in Yoreh De’ah (399:3) not to but the Magen Avraham (loc. cit.) quotes the Arizal as saying to wear them. The Mahatzis Ha-Shekel explains the Arizal’s intent being that one should wear only on Shabbos garment and the rest weekday clothing. However, the Vilna Gaon (Ma’aseh Rav, no. 193) holds that a mourner must wear his regular Shabbos garments (as did the Radbaz in his responsa, vol. 2 no. 693). The Birkei Yosef (YD 400) rules similarly.
This same dispute applies to Shabbos Hazon. According to the Rema (Orah Hayim 554:1), one should wear weekday clothes on that Shabbos like a mourner. However, the Vilna Gaon opposes this custom and it has largely fallen out of practice (cf. Arukh Ha-Shulhan, ad loc. 11; Piskei Teshuvos, ad loc. 4; Mo’adim U-Zemanim 5:343, 7:256).
The P’ri Megadim (Eshel Avraham 262:2) has a surprising leniency regarding this oblgiation. He writes: “In one’s home one may [dress] as he pleases but not in public.” In other words, you can change into weekday clothes at home but when you leave your house you shuold make sure to wear Shabbos clothes. I quote this not because it is normative – I have not seen it cited by any subsequent authority – but to show that even the most lenient posek would disapprove of the practice of many contempary Orthodox Jews. However, the Hayei Adam* writes that even when all alone in a room one should wear Shabbos clothes. The later posekim seem to be unanimous that one should remain in Shabbos clothes until after havdalah (Mishnah Berurah 262:5; Arukh Ha-Shulhan, Orah Hayim 262:3), with the exception of the Kaf Ha-Hayim (262:28) who maintains that one should nto change clothing until after melaveh malkah (and so, surprisingly, rules the Shemiras Shabbos Ke-Hilkhasah, vol. 2 42:51).
* I could not find it in that book but I found it in a smaller work of his titled Zikhru Toras Moshe 1:3.