Well, first off, what are these "non-standard" forms of English of which you speak? There are certainly unprestigious and stigmatized dialects, but each of them have their own rules of grammar. It also has absolutely nothing to do with "race". Stigmatized dialects are used by certain *regions* and *cultural groups*. For example, the Southern dialect is often considered to be "non-standard" or stigmatized, as well as dialects used by certain cultural groups such as African-American Vernacular English, and Chicano-English. Other regions and cultural groups would normally reject such forms of English saying that they were unprestigious, and stigmatized. But, even those dialects usually have covert prestige within the group using them. So, I would say that in the US it's mostly by region (e.g. the Southern dialect) and cultural group (e.g. African-American Vernacular dialect) that are considered less than "standard"--whatever that elusive standard may be, rather than by class.
Interestingly enough, however, many people that do speak stigmatized dialects, will switch to closer to the perceived "standard", in more formal situations, or when speaking to people who are not from, for example, the South, or part of the African-American cultural group, when it seems that the other person would have a difficult time understanding the dialect.