US is a superpower. Reasonable arguments can be made for Russia, China, India, Brasil, and the EU as superpowers as well. Great/major powers would include that list, plus the UK, France, Germany, and Japan.
USA is a superpower in every sense of the word, China in every economic sense of the word, not in this world are Russia, India or Brazil superpowers, the EU is not a state and in the list of Great Powers, UK, France, Germany and Japan are more powerful than Russia, India and Brazil. India and Brazil could potentially be rising superpowers but are held back by a hell of a lot of corruption, in that same sense Indonesia could become very powerful overnight if it managed to rid itself of corruption. To that sense none of the Great Powers have waged war against one another since WWII (barring a few skirmishes or post-colonial conflicts).
The war we've seen involving major powers hasn't been against other major powers; closest I can think of would be where the opposing side was supported by another major power but not actually fighting, like the Vietnam/Indochina wars (I'm not sure if that actually counts given the date), or the current clusterfuck in the Levant.
Ones that were clashes between world powers since WWII:
1950 - Korean war evolved into a major clash between S.Korea, USA, UK, PRC, USSR, N.Korea and loads more, even Turkey
1956 - Suez Crisis greatly accelerates the decline of the French and British Empires, notably the world superpowers intervened to force an end to the conflict without committing troops themselves, using financial dominance to force withdrawal without military warfare
1961 - India annexes Goa from Portugal, the war lasts a day and a half
1962 - India and China clash over their Himalayan border
1963 - Indonesian-Malayan war starts, fought between Malaya, Australia, UK, Indonesia, with support from Canada, India, USA, USSR and PRC
1965 - Indo-Pakistani war happens after India responds to Pakistani insurgents with full-scale military assault, ensuing in the largest tank battle since WWII
1967 - Six-Day War happens, in which loads of Arab armies get rekt by Israel
1967 - After the Six-Day War didn't really resolve much, the War of Attrition follows thereafter with much the same pattern and outcome
1970 - Football War happens. Though El Salvador and Honduras hardly count as Great Powers, if they have a war called the Football war it has to be mentioned
1971 - Another Indo-Pakistani war happens, this time as a result of East Pakistan becoming Bangladesh
1974 - Turkish invasion of Cyprus
1982 - Argentinians invade the Falklands, sparking the Falklands War vs UK. Notable because the world superpowers didn't step in, and the Argentinians felt proud for attacking a nuclear power - conventional rules stopped applying when Galtieri was wrongly fed information by sympathetic MPs guessing that the UK would not respond. So a very, very specific circumstance
1984 - Another Indo-Pakistani war happens, this time over Kashmir
1987 - Indo-Sino skirmish over northeastern India
1990 - Iraq war 1: Classic Gulf edition
1991 - Russo-Georgian war
2003 - Iraq war 2: The Refreedoming edition
2014 - Russia drops the donbass (with the caveat being though this is a clash of great powers, none of their troops ever clashed directly)
Some of those I had to stretch the definition to be conflicts between regional powers, everything else was pretty much proxy wars, or civil wars, or wars fought against non-state militant entities
IMO the UN has done fuckall to prevent wars, though to be fair I don't really expect anyone to be able to prevent war short of Christ descending from heaven.
I'd say the institution is irrelevant but it does have a certain air of legitimacy around it, which is better than nothing. Plus all the extra-political stuff they do is pretty useful.
Such useful extra-politically stuff being?