What is the largest town or city that has seen -40 in the 21st century? Probably International Falls.
Great question, but for sure it's not International Falls (population of ~6300). It is probably Bismark North Dakota with a population of 67,000, which hit -44 in 2009. I assume next might be Laramie Wyoming which has a population of about 32,000 and last hit -40 last January. Even Craig Colorado (where I live) has a population of about 9000, which is larger than International Falls. Even if the un-official readings at my house are discounted, the official weather station last hit -40 in February 2011 (this year the official station got fairly close with -37).
Embarrass is the coldest town in the lower 48 on average...International Falls, MN and Tower, MN...as well as Baudette and Ely, MN aren't far behind.
Several towns (tiny, but Embarrass is too-it only has a population of around 30) in the Rockies average about as cold as Embarrass (maybe slightly cooler), but they are close enough in temperature that which one is colder depends on which years are looked at.
In Colorado, Fraser, Crested Butte, Taylor Park, and Silverton are all about as cold as Embarrass and are contenders for the coldest populated place in the Lower 48 that has an official weather station, but they are all close enough in temperature that which is colder is dependent on the period of record looked at.
Stanley Idaho and Dillon Colorado aren't too far behind. West Yellowstone would be on there too, but unfortunately the weather station hasn't been operating in recent years. Leadville is up there too, but being on a slope they don't get the extreme lows that other locations experience.
Tabernash and Frisco Colorado are towns without official weather stations, but Tabernash is known to be colder than Fraser and Frisco is colder than Dillon. Breckenridge has a weather station, but it only records precipitation and snowfall. Historically it did record temperatures, not not in recent years. Still, Breckenridge is 800 feet higher than Dillon and in the same type of exposed valley location, so it is likely colder than Dillon.
A friend showed me this (obviously non-official) -52 from Tabernash when Fraser was -33 (January 2010).
Tabernash (population 417) might be the true champion of the coldest town in the Lower 48, but it doesn't have an official station.
Although they don't get the extreme low temperatures, for many years Climax Colorado would have been the coldest town in the Lower 48. The resident population mostly left in 1965, with only a very few residents sticking around after that. The mine reopened in 2012 though, so if the town does come back (which still seems unlikely), it will be the coldest town in the Lower 48 on average. Still, temperatures there (or Leadville) don't drop below the -30's. Climax actually has winter lows averaging slightly above zero, which is warmer than several other locations in the Rockies, but it is also a very windy location, so chill factors are usually lower than nearby places that get colder.
Anyway, the reason that the higher elevation towns in Colorado don't show up on the daily extremes very often is because several years ago, the NWS decided to exclude stations above 8500 feet in the daily extremes.
In winter though, Embarrass is still colder than the places mentioned above. Night-time winter temperatures at several of those other stations are comparable to Embarrass, but daytime winter temperatures in the towns in the Rockies tend to be warmer.