There are at least two that I know of: the well-known "Doctor's Plot" and the lesser "Mingrelian Affair."

The Doctor's Plot was an anti-Semitic attack on doctors, mainly Jewish, who were perceived to be trying to kill off Soviet leaders. It began around 1951, and several high level Soviet doctors were implicated. Under the direction of Lavrenti Beria, the head of the MGB (the renamed NKVD, eventually KGB), a media campaign began, trials were held, and many doctors charged and convicted. However when Stalin died in 1953 the campaign quickly ended; Khrushchev even mentioned in his 1956 "Secret Speech" that it was entirely a fabrication of Stalin, who was planning another purge.

The Mingrelian Affair was the name given to a plan to curtail the influence of Beria, who was an ethnic Mingrelian (a people closely related to Georgians; outside of Georgia they are regarded as a subgroup, but inside it is regarded as separate), and promoted other Mingrelians to higher positions within the Georgian Communist Party as a means to assert his own power over the region. Stalin was not pleased with this, seeing it as a move on him (which it probably was; Beria was setting himself up for the post-Stalin era, and needed a base of support). In 1952 Stalin brought the Georgian Politburo to Moscow and had the head replaced by his own protégé. He actually sent Beria to Georgia to further purge the Party, an ironic move considering it was largely Beria's people who were being removed. As one Russian historian wrote, "Stalin’s goal was not to destroy Beria (if he had wanted to do he would have done this), but to keep a hold on him and to remind him from time to time who is the 'boss' in the country." After all Stalin depended on Beria, but he didn't want him consolidating power while he was still alive. This was completed by the end of 1952, though how much an effect it had on Beria's future prospects I can not say.

Sources:

• Blauvelt, Timothy K. "March of the chekists: Beria’s secret police patronage network and Soviet crypto-politics," Communist and Post-Communist Studies Vol. 44 (2011), pp. 73-88.

• Montefiore, Simon Sebag, Stalin: Court of the Red Tsar, London: Phoenix, 2003.

• Rayfield, Donald, Edge of Empires: A History of Georgia, London: Reaktion, 2012.

• Rayfield, Donald, Stalin and His Hangmen: The Tyrant and Those Who Killed for Him, New York City: Random House, 2004.

• Suny, Ronald Grigor, The Making of the Georgian Nation, Second Edition, Bloomington, Indiana: University of Indiana Press, 1994.