Probably very few; keep in mind, though, that reliable sources on this are very scarce so I cannot say I'm answering this with certainty.

Anyway, I'll mention a few things I've read, so it won't be a comprehensive answer.

In the early years, during the first FYP, it is most likely that, while Stalin feared the influence of the recently expelled Trotsky (expelled from the USSR in 1929), who was publishing his Bulletin of the Opposition (Бюллетень Оппозиции) from abroad, no real plots existed. Stalin was very suspicious of any contacts between people in the USSR and Trotsky; for example, when Jacob Blumkin, one of the chiefs of the Soviet Secret Services, visited Trotsky, he was immediately shot -- Stalin wanted to make an example out of him. However, as Deutscher notes in his classic Stalin biography, 'the opposition in Russia stirred; but it did not move'. That is to say, opposition certainly existed, particularly as the first FYP put an incredible strain on society, an anti-Bolshevik emigre community was promoting its cause in the West, many Party members and Komsomoltsy were expelled and felt disenfranchised. Stalin particularly feared an emerging generation of young Communists who were less loyal to the Party leadership and more extreme in their measures: to this group of 'terrorists' belonged Nikolayev, who killed Kirov. Stalin, by 1934, was thoroughly alarmed by this threat and grew increasingly suspicious of potential plots, and this is where the purges really began. However, little evidence of real plots exists (that I'm aware of), although accusations of attempting to assassinate Stalin were commonplace at the show trials (of the sixteen, seventeen, the twenty-one as well as the secret trial of the Marshals.)

There is one possible plot that I would mention: that of the leaders of the army. This, Deutscher dubs the 'real conspiracy', and it was said to have been begun by Tukhachevsky and his associates. Possibly, civilian leaders such as Rudzutak and Mezhlauk were also involved.

In fact, little is known about this plot, and some deny its existence. In any case, there are non-Stalinist sources (such as Milyukov and Kerensky) that maintain that there was indeed a plot, from their own motives, not initiated by a foreign power (although the Menshevik Socialist Messenger also maintained it to have been an invented pretext for the elimination of Tukhachevsky). Deutscher describes it as follows:

'The main part of the coup was to be a palace revolt in the Kremlin, culminating in the assassination of Stalin. A decisive military operation outside the Kremlin, an assault on the headquarters of the G.P..U., was also prepared. Tukhachevsky was the moving spirit of the conspiracy. A man of military genius, the real modernizer of the Red Army, he was surrounded by the glory of his victories in the civil war and was the favourite of the army.'

However, the plot was allegedly discovered by the political police, a secret trial was held and Tukhachevsky was shot in 1937.

I couldn't find any authoritative accounts of this affair, and indeed there is very little hard evidence - only interpretations. At least, Deutscher considers it the one 'real' plot.

So anyway, while Stalin was deeply suspicious and indeed there were many anti-government and anti-Stalin groups and especially young terrorists like Nikolayev would have seemed dangerous, the only possibly genuine high-profile plot I could find was the Tukhachevsky one, about which much remains unclear.

Most info I got from: Isaac Deutscher - Stalin: A Political Biography

The Russian Mystery: Behind the Tukhachevsky Plot - Balticus, in Foreign Affairs, Oct 1937.

Other reading: Sarah Davies and James Harris - Stalin's World

Arch Getty and Oleg V. Naumov - The Road to Terror

Edit: I should also note the Ryutin Platform; Ryutin and his followers wrote an aggressive denunciation of Stalin's policies (first FYP), his methods and personalities, and actually called for the destruction of his dictatorship. This, Getty and Naumov note, was taken by the Stalinist leadership as 'a call for armed revolt'; it deeply troubled them, coming as it did from the upper ranks of the Party. Nonetheless, I couldn't immediately find any violent plots by the Ryutinists against Stalin.