As in any crisis situation, the crisis in Ukraine has been subject to polar interpretations. Russia, the Donbas rebels, and others are quick to paint what is happening in Kiev as a “fascist coup” and that the entire Kiev government is comprised “entirely of fascists.” On the other side, Western governments (primarily the United States and the European Union) and the Kiev government have stated that the far-right in Ukraine has marginal electoral support and thus has “virtually no influence in the country.”
As in any historical event, the truth is neither black nor white, but somewhere in between. It is indeed correct that both Right Sector (Praviy Sektor) and Svoboda are on the electoral margins (I even know people in Western Ukraine who can vouch for this). However, it would be wrong to mitigate their influence entirely.
At the end of the day, it was Right Sector and Svoboda that still played a crucial role in helping the present Kiev government rise to power. In return, they were granted positions in the government. Most of the posts awarded to the far-right have been given to Svoboda, though the newly-appointed Education Minister, Serhiy Kvit, is affiliated with Right Sector and its associated far-right paramilitary organization Trident (Tryzub). Also, Svoboda was conferred governorship of five oblasti: Lviv, Ternopil, and Rivne in the West and Zhytomyr and Poltava in the Center (though they have never been a major force in either of these two Central oblasti). Since there are a total of 24 oblasti in Ukraine, this means that Svoboda and its leader Oleh Tyahnybok effectively control about 20% of them. Further, if one were to add up the total area of the oblasti ruled by Svoboda and then divide that by the total area of Ukraine, then one would arrive at approximately the same percentage (i.e., 20% of the country overall). Given this, the question inevitably arises: how can a political party with the support of less than 2% of the entire electorate of Ukraine govern about 20% of the country?
Of the two right-wing parties, Svoboda is arguably less of a threat than Right Sector. Right Sector has very little popular support in Ukraine but they talk big and they are armed, so they cannot be easily dismissed. From my research and observations, it seems that they clearly played a role in the violence in the “anti-Terrorist operation” in the Southeastern oblasti. Right Sector is largely to blame for the Trade Unions fire-massacre in Odessa and they have also been involved in the shootings of civilians in the Donbas. Kiev has heavily relied on them and other private militias in recent weeks because their own military, drawn on recruits from all parts of Ukraine but primarily from the Central oblasti, has proven to be unreliable (with defections, etc.).
Therefore, it is indeed correct to say that the far-right has marginal support in Ukraine. However, at the same time, one cannot dismiss their potential danger, since they do retain influential political positions in the present government and their more militant segments (Right Sector, Patriots of Ukraine, Trident, etc.) are indeed armed and have participated in violence in the country.