On Shabbos in a reshus harabbim can you tie around your shoes those velcro bands with metal grips that help stop the wearer from slipping on ice and compacted snow? Yeshakuach!
This is an interesting question. In my opinion, it is permitted to wear the bands, for the following reasons.
Mishnah Berurah (301:65) discusses carrying a walking stick in the rain or the ice, where there is a concern that a person may slip and fall. Concerning this question, he mentions a dispute among poskim: some permit carrying the walking stick, and some prohibit it. For this reason, he writes that one should not carry the stick in a place where there is no eiruv.
However, the velcro band that fits on a shoe has a halachic advantage over a stick held in one’s hand, because the band is tafel to the shoe, meaning that there is room to consider it as part of the shoe itself. The reason for this is that part of the function of a shoe is to aid a person to walk without concern of slipping and falling, and the band, which gives the shoe extra grip, serves to allow the shoe to perform its function even in the snow and ice. Therefore, the band would be considered part of the shoe, and there would be no prohibition of using it on Shabbos.
In a similar sense, we find that it is permitted to walk in the public domain with an instep, meaning a pad that is placed into the shoe for comfort (Shulchan Aruch, Orach Chaim 303:15, and Mishnah Berurah 42). The reason for this is that the instep becomes part of the shoe. The same would apply to the band. Although it is placed around the shoe rather than inside it, it also becomes part of the shoe, and would be permitted.
Although the case is slightly different, many poskim permit the wearing of a nylon cover over one’s hat on Shabbos, the reason being that the nylon cover serves the hat (protecting it from the rain), and is considered part of the hat itself (see Har Tzvi, Orach Chaim 39; Beis Shlomo 39; Chelkas Yaakov 2:100; Cheishev Ha’efor 2:67; Tzitz Eliezer 10:23; Yabia Omer 5:23-24 (at length); among others). Even Iggros Moshe (Orach Chaim 1:108), who prohibits the wearing of the nylon cover, permits wearing it when the cover serves to protect one’s body from the cold and rain. This logic can possibly be extended to the velcro grip-band, which is also “worn” around the shoe (there is possibly room to distinguish from the nylon cover), and also serves to protect the body from slipping and falling.
Chazal are concerned, regarding certain items, for the possibility that a person will take something off, and carry it in the public domain. Perhaps, then, we should be concerned that somebody will take off the band, and carry it. However, because the band is worn around the shoe and treaded on constantly, it does not appear plausible that a person will take it off and carry it. Just as poskim permit the nylon cover, and are not concerned that a person takes it off (it might stop raining), so the concern would not apply to the band. However, the band should only be worn in difficult weather conditions, for otherwise the concern for taking off the band would be more plausible.
It should be added that even in a true reshus harabim–which many poskim write that we don’t have today, but which can perhaps be found in the US–many poskim would consider ‘carrying’ the band to be a melachah she’eina tzricha legofah, and only a potential rabbinic prohibition. See the above mentioned teshuva of Yabia Omer, who writes on this point at length.