It depends on the rig...some are better equipped to handle cold weather. In general, if you plan to keep your rig heated then inside pipes "should" be ok. The main worry is the basement...the water compartment, holding tanks, and supply pipes will need heat. Some rigs have a vent from the furnace to keep the basement above freezing. Others have 12vdc ceramic type heaters in the basement.
The other half of the problem is the supply point and water hose, they will need heat tape or other means to keep them from freezing. And the sewer hose should be kept as dry as possible so it doesn't freeze up.
Most rigs are not well insulated for below freezing temperatures and will use a lot of propane to keep the inside warm. Depending on how you heat the other items...if paying for electricity...you will find the electric bill will sky rocket.
If it gets real cold, near zero or below, it's almost impossible to keep the water from freezing. In our experience, low 20's to upper teen's is manageable. We've been in colder temps, but just overnight, the daytime temps warmed up enough to allow some breathing room. Much colder than that and it becomes a losing battle.
The easiest thing to do is winterize the rig and forget about using any water in the rig. Find an RV park that has good facilities and try to get a spot as close as possible to those facilities. Minimize length of stay and head south to warmer weather as soon as you can.