I've just been reading a biography of Vera Nabakov, and it turns out that she carried a gun that perhaps she was hoping to use to assassinate Trotsky. I know he was ultimately assassinated on Stalin's orders (thanks Wikipedia) but certainly the Nabakovs weren't Stalinists.
Trotsky was an individual who had the misfortune of being demonized by from both the right and the left (in this case, orthodox Communists who supported Stalin). When looking at the history of the 1917 Revolution and the early Bolshevik state, Trotsky was one of the hard-liners within the incipient Communist party. He reveled in his role as organizing force behind the Red Army and often portrayed himself as the scourge of Whites (pro-tsarist and anti-Bolshevik forces during the Russian Civil War) and Russia's bourgeoisie. This ruthlessness, both in deed and self-presentation, meant he had few friends among the Russian emigres and anti-communist forces outside the USSR.
Trotsky also had the misfortune of being the visible head of the first political faction purged by Stalin. The latter orchestrated Trotsky's downfall as a factionalist whose prewar activities (Trotsky initially was a member of the Bolshevik's prewar rivals, the Menheviks). Trotsky also alienated potential allies by acting as Lenin's heir apparent and adapting a highly arrogant and preening persona. Stalin eventually expelled Trotsky from the Communist Party and exiled him from the Soviet state. But even defanged, Trotsky presented a threat to Stalin's emerging dictatorship by arguing that Stalin perverted both Lenin and the spirit of 1917. His credentials as being the second-most visible face of the Revolution's Bolsheviks after Lenin lent weight to his critiques of the Stalinist state.
Trotskyite is an insult in several very different circles right now with very different reasons.
To Christian Conservatives he was a popular bogeyman because of his Jewish looks. He is attacked in that quality in the «A Dozen Knives in the Back of the Revolution» by Arkadiy Averchenko (1921) and in Isaak Babel's «Red Cavalry» (1922-1926) you have an antisemite character underlining that point. They regarded him as the Grey Cardinal of the Russian Revolution..
To the Stalinists he was an enemy figure. A plenty of people got killed for nothing as supposed Trotskyites during the purges which is portrayed in the Journey into the Whirlwind (1955-1967) very well. People considered too liberal would be put into that group. In Vladimir Voynovich's «Monumental Propaganda» (2000-2004) there is a short episode showing that the attitude towards Trotsky as some supernatural devil is still extant in the Russian folk.
Amongst historians there are no true believing Stalinists but there is some annoyance about his ideas about the Stalin era getting constantly brought up as factual in public. It does not have to be read up directly in Trotsky. When Hannah Arendt is discussing Soviet Totalitarianism she cites Stalin as he is cited by the Trotskyites. And Orwell's IngSoc is also mainly informed by Trotsky's critique of the Degenerate Worker's State. Historians, in general, want you to read decent history books held in high regard by themselves and not apologies of people who took part in the events.
The contemporary European left is sick to death about the sect of Tony Cliff, the european counterpart to the LaRouche sect in America. They do claim to be «Trotskyite»s, which for them is just a term inherited from the Cold War lingo for "we don't like the USSR either". They have very little in common with the actual Leon Trotsky and for the most part they just act like asshats towards everyone in disagreement with their late messiah Tony Cliff. Worst of all is that they practice «entryism» which means that if your grassroots movement or trade union or fanzine publication gains any weight you will get joined by a dozen of asshat sectarians who'll steer it away from anything it was about originally to the support of their Lord and Messiah T.Cliff. They are the scourge of all attempted at grassroot movements in Europe which is why you might find yourself told to "GTFO, Trotskyite" if you say something that sounds vaguely like the talk of that sect.