The Korean War, where the USSR fought the UN (technically speaking, but it was NATO in all but name) using China and North Korean as a proxy. The USSR sent weapons, ammunition and medical supplies to the North, as well as MiG 15 fighters and T-34 tanks.
The Vietnam War, where, again, the USSR fought America using North Vietnam as a proxy. The Soviets supplied intelligence concerning American aid raids, as well as conventional supplies as in Korea.
The Afghan War, where the USA fought the USSR using the Mujaheddin (later the Taliban) as a proxy. Most notably, they supplied Stinger missiles, which the Afghans used to great effect against the Mi-24 Hind helicopter gunships of the USSR.
These three are probably the most well known, though a few others exists, such as:
The Nicaraguan Revolution, in which the USSR supplied the FSLN, a collection of leftist political parties, and the Americans supplied the Contras, a rightist organisation opposed the the FSLN
The Greek Civil war, known as the first war of the Cold War. The USSR supplied the communist revolutionaries in Albania, Yugoslavia and Bulgaria when they rose against the British backed Greek government. The British ran out of money, so they called in the Americans. Interestingly, this affair saw the establishment of the Containment policy, which sought to curtail and contain Soviet influence.
