The Gemara in Masechet Shabbat (51) discusses the issue of crushing ice or snow on Shabbat. It emerges from the Gemara, as the Shulhan Aruch rules (Orah Haim 320:9; listen to audio recording for precise citation), that it is forbidden to crush ice or snow on Shabbat, as it resembles Sehita (squeezing). The Torah forbids squeezing grapes, for example, to produce wine on Shabbat, and the Sages enacted a prohibition against crushing snow or ice on Shabbat, which results in a liquid, because it resembles producing liquids by squeezing fruit. (The Shulhan Aruch incorporates this Halacha in Siman 320, which discusses the prohibition of Sehita.)
However, the Shulhan Aruch adds, it is permissible on Shabbat to place ice into a drink of water or wine to make it cold, even though the ice will dissolve in the cup. Since one does not actively crush the ice, but rather places it in a situation where it will naturally dissolve into a liquid, it does not resemble Sehita and the Sages therefore saw no reason to extend their enactment to include this case.
The Rishonim (Medieval Halachic scholars) debate the question of whether it is permissible on Shabbat to crush ice inside the cup, such as with a spoon or a stirrer. The Tosefta indicates that this is permissible, and this is the position accepted by the Rambam (Rabbi Moshe Maimonides, Spain-Egypt, 1135-1204). According to this view, once the ice is placed in a drink, one may stir it or crush it inside the cup. Others, however, including the Rif (Rabbi Yishak of Fez, Morocco, 1013-1103) and the Rosh (Rabbenu Asher Ben Yehiel, Germany-Spain, 1250-1327), disagree, and forbid crushing ice on Shabbat even after it has been placed in a drink. From the Shulhan Aruch’s formulation of this Halacha, it appears that he permits only placing ice in a drink and allowing it to dissolve on its own, but not actively causing it to dissolve. This is, indeed, the ruling of the Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) and of Hacham Ovadia Yosef. Therefore, while it is permissible to place ice in a drink, one may not stir the ice to accelerate its dissolution. However, the Kaf Ha’haim (Rav Yaakob Haim Sofer, Baghdad-Israel, 1970-1939) and Hacham Ovadia Yosef (in Kol Sinai) note that one may shake the cup back and forth to expedite the process of dissolving the ice, as shaking has only an indirect effect on the ice.
In light of the prohibition against actively crushing ice or snow on Shabbat, the question arises as to whether one is allowed to walk on ice or snow on Shabbat. Walking on ice or snow inevitably results in crushing it and causing liquid to flow from it. Would it thus be forbidden to walk on snowy or icy streets or sidewalks on Shabbat?
Hacham Ovadia Yosef, in his work Halichot Olam, rules that it is certainly permissible to walk on snow and ice on Shabbat. As we saw, the prohibition against crushing ice or snow on Shabbat applies only Mi’de’rabbanan – on the level of Rabbinic enactment. With regard to prohibitions in this category, Halacha allows situations of "Pesik Resheh," where a person performs an inherently permissible act that will invariably, though unintentionally, result in a forbidden act. One who walks on Shabbat has no intention to crush the snow or ice under his feet, and therefore, since this prohibition is only of Rabbinic origin, it is permissible to walk on ice or snow on Shabbat.
Summary: It is forbidden to crush ice or snow on Shabbat. One may place ice in a drink on Shabbat, but he may not stir it to accelerate the process of melting the ice. It is permissible, however, to shake the cup in order to dissolve the ice. One may walk on snow or ice on Shabbat even though he inevitably crushes it with his feet.