TL;DR: (October) It now appears that Russia is rotating its commanders in overall charge of operations in Ukraine, with the initial commander at the start of the operation still officially unclear. However, since then there have been numerous announcements of new commanders placed in charge overall.
In October, Russian Defense Ministry announced the appointment of Gen. Sergei Surovikin as overall commander in Ukraine.
I would say this is partial as others say, this information is not very forthcoming.
It is also a question that a good many others have been asking, and so I post a link to the BBC article:
Ukraine conflict: Who's in Putin's inner circle and running the war?
(Putin)... As commander in chief, ultimate responsibility for the invasion rests with him, but he has always relied on a deeply loyal entourage, many of whom also began their careers in Russia's security services. The question is who has his ear, during this most fateful moment in his presidency?
Defence Minister Sergei Shoigu
If anyone does, it is long-time confidant Sergei Shoigu, who has parroted the Putin line of demilitarising Ukraine and protecting Russia from the West's so-called military threat.
"Shoigu was supposed to be marching to Kyiv; he's minister of defence and was supposed to win it,"
He was credited with the military seizure of Crimea in 2014. He was also in charge of the GRU military intelligence agency, accused of two nerve agent poisonings - the deadly 2018 attack in Salisbury in the UK and the near-fatal attack on opposition leader Alexei Navalny in Siberia in 2020.
Chief of General Staff of the Russian Armed Forces, Valery Gerasimov
As chief of staff, it was his job to invade Ukraine and complete the job fast, and by that standard he has been found wanting.
He has played a major role in Vladimir Putin's military campaigns ever since he commanded an army in the Chechen War of 1999, and he was at the forefront of military planning for Ukraine too, overseeing military drills in Belarus last month.
"Putin cannot control every road and every battalion, and that is his role."
Russia’s Military Chief Promised Quick Victory in Ukraine, but Now Faces a Potential Quagmire
The close Putin ally modernized Russia’s forces, but those troops have run into ferocious Ukrainian resistance
The bombing of Ukraine is commanded by Russian General Surovikin
The bombing of Ukrainian cities during the Russian invasion is led by the Commander-in-Chief of the Aerospace Forces of the Russian Federation, Army General Sergei Surovikin, who personally led the column of armored vehicles during the August 1991 coup in Moscow, following the orders of the so-called GKChP.
In 1995, Surovikin, then a major, was found guilty of several crimes by the Military Court of the Moscow Garrison. The future general was accused of complicity in the acquisition and sale, as well as carrying firearms and ammunition without a permit. These articles of the then Criminal Code provided for up to 8 years in prison, but he was given only one year of probation.
It is also known that in 2014 the commander of the Eastern Military District, Colonel General Sergei Surovikin, worked in the Rostov region, where he led the dispatch of tank units subordinate to him to the southeast of Ukraine.
Surovikin was appointed Commander-in-Chief of the Aerospace Forces of the Russian Federation in November 2017. December 8, 2017 Surovikin was awarded the title of Hero of Russia for the courage and heroism shown in the performance of military duty in Syria. He was awarded the Order of St. George IV degree, Courage, "For Military Merit" and a number of medals.
In 2021, the Commander-in-Chief of the PKS of the Russian Federation, Surovikin, was awarded the rank of Army General by decree of Vladimir Putin.
Mezhidov's new video points to Kadyrov fighters' secondary role in Ukraine
Chechen special fighters continue screening localities in Ukraine, Khusein Mezhidov, the commander of the battalion "Yug" (South), has reported.
The Ukraine operation is integrated at a strategic level at the National Defense Control Center, which is inside the Russian Ministry of Defence building in central Moscow.
Russian Defence Minister Sergei Shoigu was filmed by the TASS news agency on 1 March participating in a video conference from the centre with his senior operational commanders.
Russian commanders were starting to adapt their operations and senior officers are increasingly moving up to front line areas to bring order and “impose their personality” to bring improvements to the operation. The increasing need for senior officers to visit the front has resulted in one army deputy commander, as well as a division and a regimental commander, being killed by Ukrainian fire.
Additional on how hard it is to work out the decision making process:
..there has always been a lot of guesswork in determining who might have influence over the president's decisions -- and it has only gotten harder since the pandemic due to Putin's limited personal interaction with officials.
Unlike the United States, Putin's Russia has no "regularized interagency process" in which foreign and security officials are present and making decisions. While Putin receives information and is presented with options from various individuals, it is difficult to tell "who actually influences the decision among those options,"
"It's more the last person that Putin talked to, rather than some kind of collective consensus of the foreign and security policymaking elites,"... "The whole bunker presidency just makes it even more complicated."
More about Putin's shrinking group of advisors, including Shoigu:
...Ten years ago, Putin listened to at least several dozen different kinds of people. It might have been a very strange collection of characters: at one point, it was a film director with crazy ideas about the Russian imperial past. And, at another point, it was a journalist who was a big fan of Pinochet. There were some priests. So it was a multitude of people, but now it looks like, starting in 2016, 2017, this circle has been getting smaller and smaller. And what I’m getting from my sources is that, these days, Putin listens to only three or four people. There is Minister of Defense Sergey Shoigu, whom he trusts, and that is why Shoigu has played the main role in this invasion.
Not only is the campaign being shaped by an army that has openly embraced war—...it is also being led by Shoygu, a man who has so far experienced only successes and who lacks the proper military training to understand that a battlefield victory, no matter how impressive, can sometimes lead to an even larger political defeat
The role of the military in Putin’s foreign policy
When, for instance, the Russian military attacks civilian targets in Syria, flies or sails perilously close to the aircraft or naval vessels of other states... We should instead, presume that the military is faithfully carrying out political orders.
This person has popped up:
Russia-Ukraine war: Bloody past of Colonel Mikhail Mizintsev dubbed 'Butcher of Mariupol'
The man believed to have orchestrated the devastating attack on Mariupol has been identified as Colonel-General Mikhail Mizintsev.
Officially known as the Russian National Centre for Defence Management director, the high-ranking military official has been dubbed the "Butcher of Mariupol".
Prior to the invasion of Ukraine, Col Mizintsev was also involved in orchestrating Russia's involvement during the Syrian Civil War between 2015 to 2016.
Working as the Director of the Russian National Centre for Defence – a role he's had since 2014 – it's highly likely that Col Mizintsev was a key part in orchestrating Russia's military strategy.
‘Butcher of Mariupol’ inflicts brutality he learnt in Syria -
Commander who called for officer’s ears to be cut off is blamed for strikes on schools and hospitals
Colonel General Mikhail Mizintsev, who previously headed the #Russian military operation in Syria, is personally in charge of the siege of Mariupol.
As I say this is partial, so still open.. (expecting DV's but just wanted this to be here as I hadn't seen it posted yet)
Edit 23 March 2022:
The other accepted answer has linked this today too: https://edition.cnn.com/2022/03/21/politics/us-russia-top-military-commander-ukraine-war/index.html
US unable to determine Russian commander leading war
Putin’s army leaderless, say US defence officials
In short it leads me to some thoughts:
while it is possible that Russia has quietly designated a top commander to oversee the invasion — the state of combat operations would suggest "he's inept,"
And leads me to two individuals:
Putin and Shoigu.
- Putin because he has surrounded himself with yes men and those who fear him and would rather please or appease him rather than give good, sound - if painful - advice, and this would lead to bad military decisions, for which, Putin, who is not a traditional military command leader, is not experienced in that role to do (despite FSB taking lead roles in previous small scale operations).
- Shoigu, for much of the same military-experience laxking reasons as above - he is inexperienced and not a real military commander, even more so as he is not even ex FSB. OTTOMH Shoigu is a civil engineer, not a commander rising up through the ranks of the military machine.
Whilst he has been in this position he has led a program of modernization, one that also led to changes in doctrine, often looking like (look at the reorganisation of Spetsnaz for example) he has been taking notes from Western SOF. Those changes may have worked at small scale in the past abut at this scale, the Russian war machine looks less capable.
There could be a third, but you can assume this to be someone expected to take the fall for the perceived failures in this operation.
The point is, that the Russian military, under Shoigu/Putin, changed in how it fights post Georgia in 2008. Since, the, despite the reported successes in previous small scale campaigns, the new model Russian army has never really been tested.
Those changes were not really put to the test until now. And at this ambitious size it is exposing how it is not quite ready to fight a war this way.
After all this I expect to see some scapegoating..
My personal feeling is that perhaps if they had fought it the traditional Soviet way from the start, for which they are equipped, with a real Soviet/Russian military commander in control, they might well have been in a different position than they are now.
The Russian defence minister, arguably the man most responsible for the floundering war effort in Ukraine, had not been seen in public for 12 days. Nor had the chief of the general staff of Russia’s armed forces, Valery Gerasimov.
Rumours were beginning to circulate that they may have been punished over the bungled invasion, which has failed to capture key Ukrainian cities such as Kharkiv or Kyiv and plunged Russia into economic isolation.
Some reports have also suggested that he may be under house arrest, along with others at the forefront of the fledgling invasion of Ukraine.
Update since March:
April, May, and June:
Spokesperson for the Russian MoD Igor Konashenkov stated on June 24 ...that Army General Sergei Surovikin, commander of the Russian Aerospace Forces, commands the ”southern” group of forces and oversaw the encirclement of Hirske and Zolote.
The Russian MoD’s announcement confirms ISW’s assessment from June 21 that the Russian high command is reshuffling and restructuring military command
The UK MoD confirmed that the Russian command has removed several generals from key operational roles in Ukraine, including Commander of the Airborne Forces (VDV) Colonel General Andrei Serdyukov and Commander of Russia’s Southern Military District Alexander Dvornikov, who was likely was acting as overall theatre commander.
Previously reported in April, Dvornikov, the Russian general who helped turn tide of Syrian war: (https://www.theguardian.com/world/2022/apr/10/alexander-dvornikov-russian-general-who-helped-turn-tide-of-syrian-war)
The Russian general appointed by Vladimir Putin to lead efforts to reboot the invasion of Ukraine played a prominent role in the Syrian war, where forces under his command were responsible for widespread abuses against the civilian population and were frequently accused of committing crimes against humanity.
Gen Aleksandr Dvornikov, 60, has been described as an “old school” general and a “blood and soil nationalist”, trained in Soviet military doctrines that view obliterating civilian targets as a means of gaining battlefield momentum.
It was not stated who Dvornikov replaced, if any (and if it was indeed micro-managed by Putin, they are unlikely to say so).
By June, Dvornikov was reportedly out:
General Dvornikov ‘no longer in command’ of Russian Army in Ukraine - CIT
More than a week ago, the CIT team learned from Russian soldiers that General Dvornikov was no longer in charge of "the military operation" in Ukraine. He has been replaced by General Gennady Zhidko, who is the former commander of the Eastern Military District, and now is the head of the Main Military-Political Directorate of the Russian Armed Forces.
https://odessa-journal.com/bellingcat-investigator-why-putin-doesnt-trust-general-dvornikov/ indicated that Putin had a greater influence in decision making in the conflict.
Russian Ground Forces Gen. Alexander Dvornikov, who over a 44-year military career was best-known for scorched-earth tactics in campaigns he led in Syria and Chechnya, was named overall operational commander of the war in Ukraine in April. He lasted about seven weeks before being dismissed as part of what appeared to be a wider shake-up in response to heavy losses and strategic failures.
According to Conflict Intelligence Team (CIT), an independent group that has been monitoring Russian military activities in Ukraine since 2014, Col. Gen. Gennady Zhidko took over from Dvornikov in May as the overall commander of the Russian war.
Dvornikov’s dismissal may have been linked to the destruction of the 58th Combined Arms Army, normally stationed in the southern Vladikavkaz, which was lauded as one the most combat-ready Russian armies and key to the invasion of Georgia in 2008, BBC’s Russian service reported.
But Zhidko, who also held the title of deputy defense minister, in what appears to be a damning trend for generals in Ukraine, was in charge for about a month before more problems emerged, and he was demoted to the head of the Eastern Military District.
It remained unknown who exactly was in charge of the war until Saturday (October 8), when the Defense Ministry announced the appointment of Gen. Sergei Surovikin as overall commander in Ukraine.
Russia names air force general to lead its forces in Ukraine
Russia's Defence Ministry on Saturday named Air Force General Sergei Surovikin as the overall commander of Russian forces fighting in Ukraine, Moscow's third senior military appointment in the space of a week.
The ministry did not say who, if anyone, Surovikin was replacing.
British military intelligence said in April that General Alexander Dvornikov had been appointed to take charge of Russian forces in Ukraine, almost two months after Moscow began what it calls its "special military operation", in an attempt to "centralise command and control".
However, Moscow itself has not specified that anyone is in overall military command of the operation.
Back in April, after the announcement of Dvornikov's replacement, it was noted (by CIT) that "It could be a result of a casual rotation. We saw the Russians doing the same thing in Syria".
Essentially Putin is going to rotate commanders in charge of the 'special operation' until he gets the results that he desires.