They are unable to spread through Snow Blocks. In Hardmode, the Corruption, Crimson, and Hallow can spread through it, generating Purple, Red, and Pink Ice Blocks respectively. The Ice biome is the only place where Frozen Chests ( (Old-gen console and 3DS versions) Ice Chests) can be found naturally.
Thin Ice can be found generated over water and breaks when a character falls over 4 tiles onto it (unless using the Ice Skates or another accessory crafted from them). It is composed mainly of Snow Blocks and Ice Blocks.
On the Nintendo 3DS version, the Ice biome starts at the Underground layer. The Ice biome, also known as the Underground Tundra or Underground Snow biome, is a biome that is the underground version of the Snow biome, starting at the Cavern layer and stopping at the lava layer.