The U.S. has by far the least problems with racism than any other country on multi-ethnic country on Earth. Part of this has to do with the fact that there's an actual dialogue for racism here. In the U.S., problems with racism are dealt with in the courts and in the legislatures. As such, you may think that the U.S. is a "racist" place (as Jesse Jackson would have you believe), but rather it is a sign that the system works. Most countries just deny that they even have racial problems.
Countries that are comprable to the U.S. in Western Europe have a long way to go. I lived in the UK for 7 years during the 1980's. It was not uncommon there, for people of various ethnicities to be referred to by ethnic slurs to their face and get away with it too (i.e. black people would routinely be called "awesome people", Indians would be called "Pakis", etc. - I have yet to see that happen here in the U.S., without the person who said it being instantaneously beat down). If you were Indian in the UK, you had to accept being a second class citizen. When I moved to the U.S. in 1992, one of the things that absolutely shocked the hell out of me was that white clerks working at stores would give you the same treatment as the white customers - i.e. they would smile at you, and give you the cash in your hand. In the UK, they would just sneer at you and toss the money on the table. It was as if you were doing them a favor by shopping there... Since moving to the U.S., I've been accepted as an equal by Americans since day 1. That's not to say I haven't encountered some bigots, but most people treat people of other races with respect. The UK is also a multiethnic country like the U.S. But living in the UK, I could never call myself "British", because no Englishman would ever accept me as such.
Elsewhere in the world:
In India, there are bitter divisions between the various religious and ethnic groups. Even within these groups, there are bitter divisions based on race. Politics there is almost exclusively based on playing off ethnic issues, rather than real issues.
South Africa and Zimbabwe have some of the worst problems with racism. Being a white person there today is like being Jewish during Nazi Germany. The white people living in those countries are fleeing as fast as they can back to Europe. Some of them have been living there for generations...
In Australia, there's substantial racism against Asians.
Throughout Europe, there's recently been a tide of anti-Muslim sentiment, since a several of the Sept 11 hijackers were discovered to be European Muslims. Many xenophobic, anti-immigrant parties have since come into power.
On the topic of Western European Muslims (there are now 15 million of them), most have
failed to assimilate into European society. This may be due to the fact that Europeans are extremely xenophobic and regard non-whites, and non-Christians with suspicion. This would have the effect of driving the ethnic minorities closer together and farther away from the general society. Muslims living in France, Germany, Denmark, Italy, and Spain are treated quite poorly and with considerable hostility. In contrast, most Muslims that I know who grew up in the U.S. have assimilated quite well into American society.
In Singapore, ethnic Indians and Malays have long complained of being treated as second class citizens.
In Fiji, half the population is composed of the native islanders, the other half is composed of ethnic Indians that came over in the 19th century under the British rule. The ethnic Indians are more prosperous than the native islanders, and as such the latter have considerable resentment towards the former. There was a coup last year in which the ethnic Indian President was thrown in jail, a native islander who installed himself in power, and a suspension of the civil rights of ethnic Indians living in Fiji.
There are literally countless examples. I would challenge you to find one multi-racial/multi-ethnic country in the world that has anywhere near the amount of racial harmony now enjoyed in the U.S.
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