First one would look to the various potential flash points, the USSR and PRC, India and Pakistan, North and South Korea, each armed with nuclear weapons and having past conflict and ongoing antagonism, then you have Israel and South Africa, both nuclear armed and surrounded by real and/or perceived enemies, some of whom possessed chemical or biological weaponry in response, who might use their weapons to stave off national extinction. Then one needs to consider whether the conflict and use of nuclear weapons would escalate to a global conflict.
Admittedly any of these potential regional wars even using nuclear weapons does not automatically equate to WWIII. For example if Pakistan used its limited arsenal versus India and India retaliated when a war flared up and the conventional stage was going that badly that Pakistan felt it was the option of last resort then you have a few major cities destroyed, mass casualties and loss of life, world outrage and likely little else. The USA and USSR, with the UN in the mix, would likely be working to contain this and it seems a stretch that these powers would go to blows over a regional squabble. We say this play out during Yom Kippur as both sides pressured their antagonist to back down from going nuclear or forcing that as an option. What you need is a regional war to burn out of control and necessitate super power involvement so they begin banging into each other. Think a Korea-war scenario with Soviet troops or aircraft in support, NATO on full alert as USA has its troops in combat, it only takes a misstep to get that spiraling up to full scale conflict with Lemay ticking off targets under mushroom clouds, thus the reason the USA treaded carefully and the USSR played proxy war from that time forward.
The last scenario is a rogue actor, say a nuclear Iran, and I suggest the most dangerous would be it using a bomb in a terrorist attack against the USSR. While the USA would obviously retaliate, the cold-war USSR was far less restrained, they might burn them to ash in response and if things were already heated, say at the height of the Iran-Iraq War with the western powers playing 55 Days of Peking with ships everywhere, then one might get a stumble towards the brink. Spin the scenarios from there and you might get a WWIII begun outside Europe. Good luck storming the castle!