In theory, a tornado can happen during a snowstorm, but it is not likely as they require warm and humid air colliding with a cold front. Tornadoes can and do, however, occur immediately following a snow storm or when snow is still on the ground.
Cold air funnels and winter water sprouts are also tornadoes and commonly occur during cold weather but are not associated with storms. A cold air funnel is a small tornado that can develop from a small thunderstorm. While technically a tornado, a cold air funnel does not typically cause much damage and may not even touch down, causing minimal wind damage, if any at all.
When a tornado occurs over water, it is called a water spout. A winter water sprout is sometimes called a "snownado" and is very rare. Water sprouts are funnel clouds that develop over water and, like other tornadoes, require a specific set of conditions. Winter water sprouts typically happen when cold arctic air comes in contact with relatively warm water temperatures. A recently documented winter water sprout occurred in December 2013 on Lake Superior when the air temperatures reached -7 degrees and met relatively warm water temperatures of 40 degrees.
Water sprouts can also occur in warm humid environments and happen most often in the Florida Keys.