The ICM opinion poll also suggests that half those questioned believe the government has failed to do enough to combat racism.
The survey, part of a major BBC News Online series on race relations in the UK, highlights concern in the black and Asian communities about the way the police treat people from ethnic minorities.
One in three of black respondents and 30% of Asian respondents said they have been made to feel like a criminal because of the colour of their skin.
However, more than 70% of ethnic minorities questioned - and 82% of all respondents - also said they felt the police do either a very good or a quite good job.
And asked to rate the chances of someone from an ethnic minority background ever being appointed as commissioner of the Metropolitan Police, 35% of black respondents and 39% of Asian respondents felt it was possible.
The opinion poll, weighted to include the views of all ethnic groups in the UK, is one of the largest surveys on race conducted in recent years.
The first set of results, released last week, suggested that more than half of Britons believe they live in a racist society.
The remaining findings examine views on the police and judicial process and politics.
They indicate that 37% of all those asked believe police officers discriminate on grounds of race compared to 45% who reject the suggestion.
Confidence
More than half of black respondents and 47% of Asians believe there is evidence of discrimination. Thirty-four percent of whites said the same.
And 18% of black and 15% of Asian respondents said they had experienced racism by the police or in the criminal justice system.
A Home Office spokeswoman said the department was "concerned" about the findings on police discrimination.
She said positive steps had been taken by implementing recommendations of the Stephen Lawrence inquiry and bringing the police under the jurisdiction of the new Race Relations (Amendment) Act.
This had led to changes in how the police are viewed.
She said training was being delivered which "will increase knowledge and improve the awareness and sensitivities of officers, promote best practice in policies and procedures, and improve performance in this area."
'Strive'
Commander Cressida Dick, head of the Metropolitan Police's diversity directorate, said relationships with ethnic minorities had improved significantly in recent years but said building confidence remained a high priority.
She added: "We have come a long way and continue to strive to improve our service to all of London's diverse communities."
And Gurbux Singh, chairman of the Commission for Racial Equality (CRE), acknowledged that the police had worked hard to improve their service to ethnic minority communities since the Lawrence report.
But he added: "These results show that although some progress is being made to root out discriminatory practices and offer equal opportunity to all, there is still a long way to go to improve both perception and reality."
Fred Broughton, chairman of the Police Federation, told BBC News Online the police had made "strenuous efforts" to improve race and community relations since Macpherson.
Confidence
He said: "Since perceptions and reality are not always one and the same, we would be interested to know whether the results were based on people's personal experience of the police or second hand/media reports."
The poll does suggest that the Asian community has more confidence in the police, with 47% of those asked saying they believe police treatment of people from ethnic minorities is fair.
It also indicates that black and Asian communities have more confidence in the way people from ethnic minorities are treated in court.
Asked about the government's attempts to combat racism, 50% of respondents said Labour had failed to do enough, with the results suggesting that this concern is shared across the white, black and Asian communities.
However, 35% of those asked feel that the government has improved race relations.
The survey suggests that 46% of people think someone from a black or Asian background will one day become leader of the Labour Party.
Between 7 -11 May ICM Research interviewed 1,576 people aged 18 and over.
The interviews were conducted face-to-face and quotas were used to ensure that at least 500 interviews were conducted with people from white, black and Asian backgrounds.
The data collected was then weighted to bring it into a balance with a national profile of all adults. The margin of error for the poll is plus or minus 3%.
However this margin increases when the answers are based on smaller groups within the total sample.
For example, when just Asian people are mentioned or when other individual groups are extracted from the total number of people asked.