There's a big War on the Rocks article in 2016 about just this, with numerous examples. Many of which come from RT, so directly from Russia.
To quote them, not all that much is directly aiming to support one side over the other. Rather it sows distrust:
Until recently, Western governments focused on state-to-state negotiations with Putin’s regime largely missed Russian state-to-people social media approaches. Russia’s social media campaigns seek five complementary objectives to strengthen Russia’s position over Western democracies:
Undermine citizen confidence in democratic governance;
Foment and exacerbate divisive political fractures;
Erode trust between citizens and elected officials and democratic institutions;
Popularize Russian policy agendas within foreign populations;
Create general distrust or confusion over information sources by blurring the lines between fact and fiction
For example, claiming that Clinton was sick:
The worrying photographs, circulated by major agencies, show Clinton, 68, being led up the stairs of what looks like a private home. Breitbart and other media have ventured a guess that Clinton’s well-documented brain injuries could be causing even mundane daily tasks to seem difficult at times.
RT can claim it is just repeating US news, but... Breitbart, really?
More RT stuff:
Financial propaganda weakens citizen and investor confidence in foreign markets and posits the failure of capitalist economies. Stoking fears over the national debt, attacking institutions such as the Federal Reserve, and attempts to discredit Western financial experts and business leaders are all part of this arsenal.
Let's quote RT in the linked article about the Fed:
It usually comes as a shock to people – especially diehard Americans who place infinite trust in their sacred Constitution – when they discover that the US dollar is not a product of the American government. That’s right, fellow consumers, that crumpled wad of dollars in your pocket is the product of the U.S. Federal Reserve, and despite the very official title, is about as “federal” as Federal Express. The reality is that the U.S. Federal Reserve is a profit-making venture just like Wal-Mart, General Motors or McDonald’s.
Solid journalism, that.
However, to get to the OP's question, let's just bear in mind that many cases on social media are claimed examples.
It is very difficult to actually, categorically, prove that: "yes, a viral FB post claiming XYZ" was done in Russia, by someone acting on behalf of the Russian government.
Then again, RT can be helpful. Quoting War on the Rocks
In one example from August, Disneyland Paris was the site of a reported bomb scare. Social media accounts on Twitter reported that the park had been evacuated, and several news outlets — including Russian propaganda stations RT and Sputnik — published alarming stories based on the tweets, which escalated in hysteria as the afternoon stretched on. In fact, the park had not been evacuated.
Why would a reputable news station pick up unconfirmed news from social media accounts?
Or you can track cases of fake news associated with the Internet Research Agency. From Guardian in 2020
The latest operation by the Internet Research Agency (IRA) was still in its early stages when it was detected thanks to a tip from the FBI, according to Facebook’s head of security policy, Nathaniel Gleicher. The network had 13 accounts and two pages, with about 14,000 total followers.
The Facebook accounts and pages were designed to bolster PeaceData.net, an English- and Arabic-language website that claims to be a “global news organization”, but whose editorial staff are fictitious. Headshots of PeaceData’s “staff” were created using Generative Adversarial Networks, a type of AI that can produce lifelike images of faces, according to Graphika, a social media analysis firm that produced a report on the IRA operation.
There was another incident in France last year which was attributed to Russia and aimed to spread fake mortality data about covid vaccines by enlisting influencers:
According to LinkedIn, Fazze’s management come from Moscow and have worked for an agency reportedly founded by a Russian entrepreneur.
French media have pointed to the similarities between Fazze’s message and the official Twitter account of Russia’s Sputnik V – a viral vector vaccine like AstraZeneca – which has repeatedly claimed “real world data” shows they are “safer and more efficient” than mRNA vaccines.
And, getting back to elections, in September 2020, it seemed that Russia was manipulating BLM coverage in social media (both to discredit it and to inflame it - again it's more about division than pushing a winner):
Going forward, Russian-linked trolls could exploit ongoing reports of law enforcement brutality against protesters to further amplify division. The Kremlin’s previous attempts to hijack movements such as Occupy Wall Street and the Standing Rock protests against the Dakota Access pipeline indicate that the Kremlin can skillfully reframe social protests to increase mistrust between U.S. citizens and their government.
If African Americans continue to be the main target, we would expect the Kremlin to increase efforts to shape race-related debates leading up to and during Election Day. Without robust online safety frameworks, it will be difficult to insulate the legacy of recent protests from Russian interference.
So, to recapitulate, there are lots of claims about social media faking by Russian actors but it is hard to prove. The smoking gun is more present when official Russian news posts stuff themselves but then again Western media is hardly always factual (Breitbart) and even solid news organizations have little good to say about Russia these days. Make no mistake, as you read the news: a lot of it is written to promote the Western position. Which doesn't mean it's false, only that you need to be careful in choosing what to believe at face value.
Ask yourself however, after the events of the last few months, how you would evaluate the credibility of the various governments?
