There are number of cases of Soviet illegals in the United States. However they were not really the sleeper agents of The Americans. They were generally used as couriers between the KGB legal representatives in embassies and their agents in US government agencies.

Rudolph Abel, as an example, entered the US in 1948 to reactivate the nuclear spies who had supplied the USSR with information on Los Alamos during the war. Abel organised a network which included the Rosenberg's who couriered information on US nuclear research.

In 1953 Reino Hayhanen, Abel's assistant, accidentally spent a hollowed out nickel, which after months in circulation, was broken open by a paperboy, revealing the micro-photos inside. This alerted the FBI to the presence of spies, but they could not break the code. Eventually Abel became tired of Hayhanen's alcoholism, had had Moscow summon him home, on his way back, in Paris, Hayhanen defected to the US, leading to Abel's capture.

I think this shows where the Americans falls down, in that illegal agents generally acted as conduits for informarion from agents in place to the KGB. The antics of the family in the Americans, with murder and assassination are improbable to say the least

Having said that, Mitrohkin and Andrew, in the Sword and the Shield, allege that comprehensive sabotage was planned in the event of war (which likely inspired at least one episode of The Americans). They allege there were both safehouses and caches ready for a campaign of sabotage on oil refineries, power stations, and more. Mitrohkin was a defector from the KGB, and his claims have substantial basis in fact, as caches have been found in Switzerland and around Europe, even Canada

Edit: In short, there are two elements to my answer; confirmed sleeper agents were generally lived rather dull lives as couriers, so far as we know, at least in the US (Trotsky was after all assassinated in Mexico). Certainly with a lack of murder and mayhem. Secondly, it seems very likely the USSR had plans to conduct widespread infiltration of the US, and sabotage of high value targets, in the style seen in the Americans. This probably involved weapons caches and safehouses, so (wild speculation) sleeper agents might have been involved, but only in the organising, as any action would have been a job for the Spetsnaz.

The show is certainly based, at least to some extent, in fact. There were spies for the Soviet Union living as "regular," people in the United States. Arguably the most famous case is the Rosenbergs who were executed after being found guilty of conspiracy to commit espionage, specifically gaining information about the American atomic bomb program. While the Rosenbergs are the most known (almost certainly because of President Eisenhower's decision to execute Ethel Rosenberg) there were numerous other Atomic Spies during the beginning of the Cold War. Note that this list includes more than just Soviet spies in the United States.

Regardless of your opinion of Joe McCarthy (I personally don't care for the man,) it can be argued that his Red Scare hampered KGB attempts to regain a foothold behind the scenes after the Atomic Spies were rooted out. However, they did still have victories. The largest victory lasting for almost 20 years in the form of John Anthony Walker who gave the Soviets incredible access to the movements of the United States navy.