There were numerous "close calls" to the use of nuclear weapons. The Berlin airlift crisis, for example, as early as 1948. The Korean War. The Cuban Missle Crisis. Many more. Some of these came quite close to actual use, especially in a tactical roles by soldiers who felt they were at risk and has access to nuclear arms. In general, this kind of big nuclear strike was avoided in part because of the threat of retaliation.

In most cases it is easy to understand why the Soviets would not have been too tempted, at a political level, to use these weapons -- they were often quite out-gunned, surrounded, and were nearly guaranteed to suffer disasterous consequences, with little hope of getting much positive out of such use. The more interesting case is that of the United States, esp. in the earlier Cold War when there was nothing like nuclear parity between it and the USSR. With Berlin, it was not something the US would want to do unless it really did want to start another general war, one that would not yet look like a "24 hour war," but would be a version of World War II with the occasional destruction of cities. It would still require huge manpower and force expenditures and a troublesome question of what to do next. This was not an appealing idea for Truman or his advisors.

Why didn't the US use atomic bombs in Korea, when the USSR lacked the practical ability to threaten the US in kind? The use of them was advocated by high-level military officials (including MacArthur) but vetoed by the President. Here we see a different set of considerations at play -- concerns about world opinion, concerns about the way atomic bombs should be considered as weapons (did we want to open the door to the USSR feeling that it was acceptable to use these weapons tactically against our allies?). In the end, the US came remarkably close to potentially using them -- Truman authorized sending bombs and bombers to carriers in the region -- but did not.

The creation and maintenance of the "nuclear taboo" was by no means obvious or destined. In the end, the persistent conclusion was that it was in the US interest to not "normalize" the use of nuclear weapons, even in a "tactical" situation (that is, not targeting cities), because it would ultimately set up precedents that would harm the long-term military interests of the US, and do great damage to arguments of moral superiority which were necessary for enrolling allies against the Soviets. This is separate from the issue of retaliation and deterrence and Mutually Assured Destruction, which only came into play a bit later in time. (By the 1960s the Soviets could probably have done a lot of damage to major US cities with their missiles but it is not really until the 1970s and beyond that they really coulda inflict an equal amount of damage to the US reliably as the US could inflict onto them.)

Nina Tannenwald's The Nuclear Taboo is a systematic study of the creation and maintenance of the policy of the non-use of nuclear weapons, even in non-MAD situations, and the best resource on this topic that I know of.

The Battle of Khe Sanh during the Vietnam was one one point where the use of tactical (rather than strategic) nuclear weapons was considered. As the condition on and inside the base perimeter worsened during the Tet offensive the Joint Chiefs of Staff sought approval for the use of chemical and nuclear weapons on the besieging North Vietnamese troops surrounding the outpost. However, this plan was shelved:

"Because of terrain and other conditions peculiar to our operations in South Vietnam, it is inconceivable that the use of nuclear weapons would be recommended there against either Viet Cong or North Vietnamese forces.

Robert McNamara, signed"

Tactical nuclear weapons would of been of dubious effectiveness in the broken, hilly terrain surrounding Khe Sanh, and the situation never worsened sufficently to warrant lethal chemical agents being deployed. The sheer devastation wrought on the hills surrounding Khe Sanh made nuclear weapons unnecessary anyway, as the amount of ordnance dropped on Khe Sanh was actually equivalent to several Hiroshima sized nuclear weapons.

There were lots of close calls and periods of increased tensions where there were some figures who advocated preemptive strikes.

-The Berlin Airlift crisis had a real fear that the Soviets would make a conventional push westward into West Berlin and West Germany. Soviet tanks and conventional forces at the time far outnumbered those of NATO, so any counterstrike would likely have been nuclear. It's hard to say how much the presence of nukes played into the decision of both sides to keep things peaceful, as they still weren't very numerous in this era. Just the threat of conventional war was probably enough to deter both sides, with the memory of WWII still so fresh.

-The Korean War, where MacArthur advocated using nukes on China, but was overruled due to not wanting to escalate and make usage of nuclear weapons the norm.

-The Cuban Missile Crisis put the US at Defcon 2. The Joint Chiefs advocated a full scale attack and invasion of Cuba that probably would have involved nuclear weapons. Castro wanted to conduct a preemptive strike against the US and pushed the Soviets for one. A Soviet submarine was depth-charged by the US blockade forces and had an argument break out on whether to go up for air or use a nuclear torpedo.

-The 1983 Able Archer exercise included a simulated escalation to Defcon 1. This came at a time when the Soviets were focusing on watching the movement and behavior of NATO leaders to give extra warning of an impending attack, when they thought that Reagan was a madman planning to cover a preemptive strike with his "Star Wars" initiative, and when the US was engaging in psychological warfare exercises where they would send naval and air strike forces charging at the Soviet border only to turn away at the last moment, so as to gauge the response. I've seen some analyses that argued that this was the closest we came to nuclear war.

-Numerous accidents and mistaken reports: continuing with 1983, in the midst of the crisis a Soviet satellite detected nuclear launches in the US. The operator decided it was a false alarm, later determined to be caused by sunlight shining on clouds. In 1995, Russian radar stations picked up a rocket heading on a high altitude course that appeared to be from North Dakota to Moscow. It fit the profile of a preliminary EMP attack. The Russian nuclear football was actually activated and presented to Boris Yeltzin, the only time any country has done so. It was a Norwegian research satellite, whose launch the Russians had been properly informed about. On the US side, at the height of the Cuban missile crisis, an airgroup in Okinawa reportedly received the go code for a nuclear strike on China during their routine check-in. They hesitated until they could get confirmation due to their still only being on Defcon 2 and the strike being targeted on China, not the USSR. Later, in 1979, NORAD recieved an alert that the USSR had launched a full Nuclear assault. Interceptors and the President's emergency plane were all put in the air. It was later determined that a technician had accidentally run a training program.

There are more IIRC, but that's a lot of the major ones.