Since 1990, the General Social Survey has asked "What do you think the chances are these days that a white person won't get a job or promotion while an equally or less qualified black person gets one instead? Is this very likely, somewhat likely, or not very likely to happen these days?" Although the GSS evaluates proposed questions carefully, they missed a problem of interpretation here: people can evaluate the hiring of "equally" and "less" qualified blacks very differently. A person who believes in that hiring should be according to merit would believe that if two people are equally qualified the choice should be made at random, or on a first-come-first-served basis. That is, if some equally qualified blacks are getting chosen over whites, that's justice; if less qualified blacks are chosen over whites, that's injustice.
However, I think that people focused on the "less qualified" part: the GSS had some questions about why blacks generally earned less than whites, and people who said that the "equally or less qualified black person" was likely to get a job were more likely to say it was because of lack of will power or inborn ability and less likely to say it was because of discrimination. So this question can be taken as a measure of "resentment": belief that blacks are getting unfair advantages. (Logically, someone might say that a less qualified black person is likely to get a job and that's a good thing--that is, they might accept the "classic" argument for affirmative action and think we have a lot of it--but it seems safe to say that's a minority position).
The figure shows the trends in opinions for whites and blacks: higher numbers on the vertical axis mean that people on the average think it's more likely. Whites have moved pretty steadily away from thinking that it's likely (except maybe between 1990 and 1994). Blacks have become more inclined to think that it's likely. As a result, the differences in opinions between blacks and whites have almost disappeared. So this question suggests a clear decline in "racial resentment" over the last 20-25 years.