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I’m in Chicago, and I grew up in Iowa (which gets just as cold). It’s not the outside spigot that is being talked about. It’s inside.
The problem is that indoor pipes often run through insufficiently-heated areas. Some builders of attached-garage homes run pipes through the garage. Or just pipes in an outside wall that isn’t insulated well enough. On very cold days (especially with wind), the inside of the wall can get below freezing.
There’s one part of the equation that SDSTAFF Ken didn’t mention. Depending on how far North you live, there is a depth in the ground that is (I think) called the frost line. It is the deepest point that the ground freezes hard during winter. Underground water pipes are buried well-below the frost line, meaning that the dirt surrounding those pipes never gets below freezing. (That is also why deep caves stay a fairly constant temp during both summer and winter.)
Thus, if you keep your water dribbling, you are constantly bringing fresh above-freezing under-ground water into a pipe that is exposed to below-freezing air. But, depending on the length of pipe, and how cold the unheated area is, it is a race against time. The warmer water in the pipe cools as it flows. If it is flowing too slowly, it will reach freezing temperature before it has reached the end of the unheated run of pipe. So, the colder the weather, the faster you have to let the water run to prevent freezing.
Just insulating the pipes better just gives you more time. Maybe you only use a particular sink for brushing your teeth in the morning. If the pipe is well-enough insulated, it can slow down the cooling process to the point that it only needs a once-a-day hit of fresh water. (Just be sure to run the water long enough to completely replace the water in the pipe … a good idea even in the summer to minimize impurities.)
But if the ambient temperature around the pipe stays below freezing, it doesn’t matter how well you insulate the pipe, it will eventually freeze if no fresh water is allowed to enter. (E.g. if you are on vacation in the Bahamas during a week of sub-zero temps at home.)
Here’s a silly invention idea. Have a device that measures the temperature of the water coming out. As the water temp approaches 32 degrees, the device increases the water flow. As the water temp rises, decrease the flow. This would let you minimize the impact on your water bill.