In my view, this is the wrong question to be asking. There can be no question that racism still exists today. It is very likely that discrimination still exists. The more important question is whether racism and discrimination continue to be major problems that continue to hold non-white people back.
I do not think that it is possible to deny that racism continues to exist in the United States today. For example, in the link below we can see that police officers and court officials in Ferguson, Missouri thought that it was appropriate to send around emails with racist statements and images. We have also seen the example of the fraternity at the University of Oklahoma whose members were videotaped singing songs about lynching black people and about how they would not want blacks in their fraternity. There is also racism against people from many other groups. In fact, there are even people of color who have racist attitudes about whites. Racism clearly still exists.
It is harder to prove that discrimination still exists because hardly anyone is stupid enough to say that they are, for example, refusing to hire someone on the basis of race. However, studies have been done that show that when identical resumes are sent out, those with stereotypically black names get fewer calls for interviews than those with names that sound white. This suggests that discrimination still occurs.
The bigger issue is how widespread racism and discrimination are and how much they affect their victims. Are racist attitudes serious impediments to minorities’ efforts to get ahead or are they just personal opinions on the part of a few people that do not end up hurting anyone in any important ways? This is a much harder question to answer and one that is much more important for our society.