The Shulhan Aruch permits walking on snow on Shabbat, even though doing so may melt the snow into water. The Aruch HaShulhan (Rav Yechiel Michel Epstein of Nevarduk, 1829-1908) explains that even though the Shulhan Aruch prohibits rubbing snow with one’s hands, as a Gezerah against Sehita (Squeezing), the Hachamim did not institute this Gezerah against walking, because it would be too difficult, especially in snowy areas, for a person to remain inside lest he tread on snow on Shabbat. The Ben Ish Hai adds that if there is a way to avoid walking on the snow, on should be strict and take the detour. He concedes that if there is no alternate route, it is permitted to walk on the snow, even if it is already thawing and treading on it will certainly turn into water.
The Poskim discuss whether it is a problem to walk on snow with shoes that have an imprint on the sole, such as a Nike logo. Hacham Ovadia rules that there is no problem, as he certainly has no intent to make a design on the snow, and such an impression is not considered writing from the Torah anyway.
MaHaram M’Rutenberg (Rabbi Meir of Rothenberg, 1220-1293 Germany) permitted urinating in the snow, even though it definitely melts the snow. The Hachamim did not prohibit this, as it is a vital bodily function, especially in times when there was no indoor plumbing. Although the Rosh (Rabbenu Asher Ben Yehiel, 1250-1327) himself did avoid doing so, Hacham Ovadia explains that he did not prohibit it for others.
It is permitted to walk on snow or to urinate on snow, even if doing so causes it to melt. There is no problem of walking in snow with shoes containing an imprinted word or logo on their soles.