The Sages forbade crushing snow, hail, or ice in order to turn them into water, because this resembles a melakha (see the next paragraph). However, one may add ice cubes to water even though they will melt. Since he is not actively melting the ice, it is not prohibited (Shabbat 51b). One may chip away at a chunk of ice in order to put some into a cup or container, even though bits of ice may melt during the process, as it is not his purpose to turn the ice into water. Similarly, one may walk on snow even though as one walks a little snow will melt, because this is not his intention (SA 320:9-12).
A few poskim (Sefer Ha-Teruma and Rosh) maintain that one may not crush ice and turn it into water because turning a solid into a liquid is considered producing something new. According to this, heating congealed gravy or the like is also prohibited, since heating the gravy will liquefy it. However, according to the majority of poskim, the prohibition of crushing snow is not because something new is being formed, but because crushing snow is similar to squeezing fruit in that both actions are done with one’s hands. According to them, one may liquefy congealed gravy indirectly by warming it (Rambam, Ramban, Rashba, Smag, Smak). The Sephardic custom is to be lenient in this regard. The Ashkenazic custom is to be stringent le-khatĥila and refrain from liquefying congealed food by heating it. When necessary, even Ashkenazim may be lenient (SA 318:16).
This is the law for making ice as well; Sephardic tradition allows it, while some Ashkenazim are stringent le-khatĥila. However, when necessary, such as on a hot day, even Ashkenazim may make ice (SSK 10:4). Some Ashkenazic poskim are lenient even le-khatĥila and allow making ice even if it is not a hot day. This is because ice does not have a sustained existence as a solid. It starts melting immediately upon being removed from the freezer, so making ice is not really creating anything new (Tzitz Eliezer 6:34; 8:12).
One may add spices to foods even though they create a new smell in the foods. Although one may not cause clothing to smell nice by creating a new smell, there is no comparable prohibition for foods (SA 511:4; MB ad loc. 24).
According to many poskim, one may not whip an egg or make whipped cream, because it looks like one is preparing them for cooking (Shabbat 109a following Rashi; MB 321:68). Others are lenient (see Livyat Ĥen §66).