The concept that just because a nuclear device is detonated suddenly missile silos are all firing off like crazy everywhere is actually outdated, there are many options on the table that are all situationally dependent. The most likely cause for nuclear war is that two nuclear powers are engaged in conventional warfare and one begins losing and deploys a nuclear weapon against a military target. After this happens we have several options:
Option 1: Call their bluff. They aren't REALLY prepared to commit to a nuclear war, this is a scare tactic. Rally on the moral high ground that you will not use nukes and press the attack on the ground.
Option 2: tit-for-tat reciprocation. We signal that we are perfectly willing to enter into a nuclear war but would prefer not to by striking a single military target of equal value to the one we lost. The hope is that they realize the gravity of what they are escalating into and decline to escalate the situation further.
Option 3: Escalation. We destroy 2 targets of considerably higher value than the one they did. We are signalling that we aren't just ready to enter into a nuclear exchange, but we are willing to be the ones who escalate it.
Option 4: limited exchange. We retaliate against multiple targets avoiding civilian population centers and instead focusing on missile launch sites, bunkers, and military installations. We are hoping to destroy or disable enough of their ability to retaliate in one blow with minimal casualties because we believe their successful strike may have emboldened them into attempting to do the same to us.
Option 5: All out warfare. We fire everything we have at any target we can because we believe that now that our enemy has opened pandoras atomic box there is no going back.
A very major part of it is psychology, you don't always need to obliterate an enemy to defeat them. So the question "how many nukes to defeat a major superpower" is highly culturally and situationally dependent.
The USA has historically been a very, very poor nation to attack. The Lusitania, Pearl Harbor, and 9/11 have historically demonstrated that trying to psychologically sucker-punch the USA into an early defeat backfires spectacularly. The USA tends to actually GAIN morale from such events, repackaging a disastrous defeat or a horrific disaster into righteous fury and quickly mobilizes its culture for all out-war. The problem with the USA is that they grow tired of war equally quickly. The Tet Offensive in Vietnam or the Chinese offensive at chosen reservoir in Korea demonstrate how even a losing battle fought at the right time can shock and demoralize the war-weary American people into agreeing to a cease-fire or withdrawing early. A nuclear strike against the us right when the end of the war was in sight could very effectively break US morale just like the Tet offensive did in Vietnam.
Russia is a very different beast, they have always been stoic line-holders. The psychology of how they operate is reflected in a "retreat into winter, allow the enemy to exhaust themselves, counter attack when we've weathered the storm" mentality that is heavily ingrained to their culture. The Russians also rule more along oligarchical lines than the USA and components of the military are seen to a large part as expendable so long as they achieve some goal in preserving the momentum of the whole. Soviets tend to have a much more defensive mindset. You'd have to pound them pretty hard to break them. Look at Stalingrad, they lost almost half a million men fighting for a city that contained nothing of value except their leaders name. They were more than willing to sacrifice half a million men simply to create an expensive tar-trap for the nazis. You would have to launch a a decent sized series of strikes crippling their military to break the Russian stoicism.
The Chinese are a bit of a wild-card. The Majority of engagements they fight seem to be civil wars. The long march, and the hundred flowers purge shows that there are really two cultures in china. The inner party and everyone else. Everyone else are expendable in numbers the Russians would even find appalling. China's military is structured to match, despite all what you hear about their hundreds of millions of soldiers, there really two militaries in china. The first is the rank and file conscripts, the 2nd are the elite inner party controlled units. These inner party units are dedicated mostly towards preventing the non-party assets from deserting or rebelling, and that's during peace time. To defeat china you would need to launch a decapitating strike and hopefully without the inner-party's leaders their control over the elite units would collapse. Without guns at their backs the rank and file non-party military would surrender or desert.
These are all approximations and hypothetical meant more to demonstrate the differences of response. The reality is we've never tested nuclear tactics in battle so any talk of nuclear war is going to remain thankfully theoretical.