Language is such an integral part of a cultural identity, so intimately linked to who we are, that cultural conflict over dialect and the languages of immigrants has always been a source of tension.
The first waves of Irish immigrants in the 1830s and 40s met with political and social backlash, even though they spoke English, because Americans saw the Irish brogue as something culturally and religiously (Catholic) threatening. The reaction has been similar to every immigrant wave since that time.
Spanglish is not only controversial with English speakers, who see it as an assault on the native tongue they have always spoken, but among Spanish speakers who also see it as a threat to their heritage, with a fear of losing their identity in the English culture they are surrounded by.
Ebonics, or Urban English, also known as African-American Vernacular English, represents a different controversy. The dialect is part of African-American culture, born in slavery and discrimination and developed over the generations as unique to their population. Thus, it is common to feel resentment and resistance when pressured to speak a different dialect of English, especially one common among whites who for so long denied them their cultural and linguistic identity.
Also included in this debate could be the "southern drawl" and the stereotypes Americans have come to associate with it. Ever notice how none of the major mainstream media outlets use news anchors with a southern accent?