This is a relatively difficult question and is very close to falling within the 20-years rule. It might have felt like ages, but the USSR fell not very long ago. If you look on this wikipedia page, you will find a list of known Soviet agents in the US. You will not a find a lot of "spies" beyond the 1970's. Moreover, the greatest part of named spies were in the 1940's.

However, the real kicker here is that many of these spies were American nationals. There were seemingly very little Russians in the USA posing as Americans. That is logical as well if you think about it, it's a lot easier to convince American nationals to feed you information or operate on your behalf than to make Russians nationals immigrate to the US in a period like the Cold War and pretend they're Americans.

That does not mean that there weren't any attempts to, however. Some evidence here is breaking the 20-year-rule, but that's just the danger of the question posed. In june 2010 the US arrested 11 Russian agents and deported them. They had been there for over 10 years, see this NY Times article, which nicely includes a quote from former KGB General Oleg Kalugin saying:

“It’s a return to the old days, but even in the worst years of the cold war, I think there were no more than 10 illegals in the U.S., probably fewer.”

Whether he is a trustworthy source is difficult to say, but at this moment there seems to be no reason to assume he would lie.

In short, the Cold War spying of highly trained spies turns out to be the same old boring story; informants do the real work, sleepers and spies are overly exaggerated in popular culture.