There are several deaths in the novella. Which is most revealing and why? What death (including the mouse's) shows us about life back in the 1930's?

In Of Mice and Men, I think that the death of Candy's dog illustrates a great deal about the 1930s. 

The struggle to find work is significant to the time period in which Of Mice and Men takes place. The time period was filled with people who needed work and would move from place to place, searching for it.  People had to be "of use" in order to find such work.  If individuals were not of use, they were simply discarded.  The economic hardship that defined the time period rooted out those who were not useful, relegating them off to the margins to be forgotten.

Candy's dog used to be productive.  Candy describes him as the "best damn sheep dog."  There was a time when the dog used to contribute in an active and productive manner.  However, age has taken its toll on the dog and he now is a shell of what he once used to be.  Carlson is the most active in pointing out the uselessness of the dog.  He says the dog "don't enjoy nothin'" and how "He ain’t no good to you, Candy. An’ he ain’t no good to himself."  Even Slim says that Candy's dog lacks a utilitarian purpose:  "Carl’s right, Candy. That dog ain’t no good to himself. I wisht somebody’d shoot me if I get old an’ a cripple.”  In the world of the 1930s, when one loses their productivity and ability to work, they are of no use to anyone.

Candy's dog is taken outside and shot. His death is away from everyone else. Candy is the only one who remembers him.  Steinbeck writes of "the silence" that everyone in the bunkhouse experiences, but that only Candy feels. When Candy's dog is shot, it reminds us that the 1930s defined a person's ability to work as the most important reason for being in the world.  As seen with Candy's dog, when they lose this ability, they become expendable.