I think we need to define socialism before discussing the finer points of it. I think a textbook definition is entirely appropriate because everyone probably has a somewhat different conception of socialism. All though socialism has been implemented in various ways previously, this is a theoretical discussion, so theoretical socialism should be discussed.
In theory, socialism is a method of controlling the economy advocating state, public, or common worker ownership (My commentary: Ultimately that means government/people/common workers control everything from prices on goods to which goods get produced.) and a society characterized by equal access to all resources, meaning a more egalitarian - economically equal - society. (My commentary: all though this could mean that everyone gets a Ferrari, more commonly this equality has been achieved by limiting the quality of goods due to costs of production, etc.)
Socialism is not a government system. The citizens of the United States - if they so desired - could inform their representatives that they wanted a socialist economy, and the representatives could legislate that, and then they could directly control each and every businesses operations. (I don’t know whether there are any constitutional conflicts or not. Of course a precedent for government owned business in the United States has been set with the United States Postal Service.)
Ripped straight from wikipedia: Socialists mainly share the belief that capitalism unfairly concentrates power and wealth among a small segment of society that controls capital, creates an unequal society, does not provide equal opportunities for everyone to maximize their potentialities in society and does not utilize technology and resources to their maximum potential in the public interest. Therefore socialists advocate the creation of a society in which wealth and power are distributed more evenly based on the amount of work expended in production, although there is considerable disagreement among socialists over how and to what extent this could be achieved.
The people who would probably support socialism are common workers, democratic societies (the textbook definition of democracy, not republicanism like in America), dictators, and technocrats (people who believe that the academia are most qualified to govern).