I agree with Riain's point.
The 1973 war in the Middle East could have triggered WWIII. IIRC the US was on a very high level nuclear alert, complete with B-52 "orbiting" in northern Canada, and there were fears of Soviet intervention to defend the Syrians from the Israelis, who were threatening Damascus itself.
Something getting squirrelly in South Africa could be a flashpoint. Say there's a Soviet-backed attempt by the black states bordering South Africa to drive the white people into the sea. The South Africans threaten to use their nukes if the West doesn't intervene, the West refuses, and the South Africans unleash their nukes.
The Soviets, betting on world outrage at the South Africans' behavior (on top of the usual anti-apartheid sentiment), get involved in the war ostensibly to put down a "rogue" state and the US intervenes to ensure the Soviets and their pawns don't take control of the Cape route.
There's a novel called The Sixth Battle involving a war between the US and some kind of reformed Soviet-ish state over South Africa. The Soviets have CBGs of their own, so fun ensues.