I found an interesting thread googling M37. http://www.shtfblog.com/shtf-blogs-top-ten-best-guns-for-survival/
Having been on the other side of the world, and done some time in a combat zone, I like to think that I have a somewhat educated opinion of what "anarchy" would look like.
My guess is that "TEOTWAWKI" is going to be more Iraq and less Somalia.
First you will have the event that causes the fall, man made or natural doesn't matter.
Then you will have people who start organizing things. In Iraq it was tribal leaders and clerics, in Somalia it was "warlords".
In Iraq you had an intentional stabilizing force (the Coalition) and intentional destabilizing forces (terrorist factions of at least three major flavors). In the end it was the people themselves who decided whether the nation of Iraq would descend into Somalia or start on the road to joining the civilized world. I figure that we will have the same options.
What Iraq (and the US) have going for it is a very high literacy rate. What the US has going for it more than Iraq is a lack of tribal or religious loyalties amongst the population. Sure here we have the Amish and the Mormons, but all in all neither the Amish nor the Mormons (or Mennonites, Hudderites, Catholics or Baptists) are going to shoot you for not going to church with them. Elsewhere in the world that is not the case.
Americans have this amazing ability to pull together in times of crisis. Our literacy rate means we don't lose laws, they stay on the books and we have our tradition of rule of law (not rule of force). If a nuclear warhead took out Washington DC and eliminated both houses of Congress, Pres, VP, and all members of the Supreme Court, the rest of us would shed a few tears and go about setting up elections to reform our government.
A disaster doesn't destroy the rule of law if the people respect the rule of law. Heck, when the Polish president went down Poland didn't have generals pulling a coup to fill the power vacuum.
It would take something very very serious, like multiple nukes or virulent bioweapon to shake this country.