It is commonly assumed that one may wear eyeglasses outdoors on Shabbat, even in an area that is not surrounded by an Erub, and that this does not constitute "carrying" on Shabbat. Interestingly enough, however, the Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933) wrote that it is forbidden to wear eyeglasses in a place without an Erub on Shabbat. It seems that the Mishna Berura was referring to different kinds of eyeglasses than the ones normally worn today. He spoke of lenses in a frame without earpieces, which just sat on the nose. Because these lenses were not securely fastened to the face, the concern arose that one may remove them, and the Mishna Berura therefore forbade wearing them in a public domain on Shabbat. Today’s eyeglasses, of course, have earpieces that ensure that they are worn securely on the face, and there is thus no concern that one may remove them. Therefore, according to the vast majority of recent and contemporary Poskim, it is permissible to wear eyeglasses in a public domain on Shabbat, though it is, of course, forbidden to remove them in a place without an Erub.

Hacham Ovadia Yosef similarly allows wearing sunglasses outdoors on Shabbat, as we are not concerned that one will remove them. Once again, it is certainly forbidden to remove sunglasses on Shabbat in a place without an Erub.

Is it permissible to wear Galoshes over one’s shoes in a public domain on Shabbat?

Rav Moshe Feinstein (Russia-New York, 1895-1986) addresses this question in his Iggerot Moshe, and rules that it is permissible to wear Galoshes in a place without an Erub on Shabbat. He explains that although Galoshes are worn over one’s shoes, they are nevertheless considered a garment, since they provide additional warmth and prevent the feet from getting wet. Therefore, they may be worn on Shabbat in a public domain, just as Halacha permits wearing any garment in a public domain on Shabbat, and does not consider a person to be "carrying" the garment. However, this applies only to fitted Galoshes. If the Galoshes are not properly fitted, then one might remove and carry them in the public domain, in violation of Shabbat. Fitted goulashes, however, may be worn on Shabbat, as discussed. This is also the ruling of the Aruch Ha’shulhan (Rav Yechiel Michel Epstein of Nevarduk, 1829-1908).

Summary: It is permissible to wear eyeglasses and sunglasses outdoors on Shabbat in a place without an Erub, unless the eyeglasses do not have earpieces and simply rest on the nose. Of course, one may not remove one’s glasses outdoors on Shabbat in a place without an Erub. One may wear fitted Galoshes outdoors on Shabbat; Galoshes that are not fitted should not be worn outdoors on Shabbat in a place without an Erub.