This isn't exactly the newest of news stories, but sometimes these things need to be repeated, simply to keep pointing out how major world superpowers should not be taking tips from comic book supervillains (or perhaps providing inspiration for them).
Back in the late 1950s, when the United States and Soviet Union were really getting the whole Cold War machine churning, someone in the U.S. Military came up with a secret project called "A Study of Lunar Research Flights". In the growing escalation of technological one-upsmanship between the two superpowers, this was to be the one to beat all others, to convince the Soviet Union to tuck its tail between its legs and just give up — the U.S. was going to nuke the Moon.
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The plan, nicknamed 'Project A119', was to launch an intercontinental ballistic missile at the Moon, targeted specifically at the the Moon's terminator — the line between the light side and the dark side. Timed properly, the immense mushroom cloud produced would be fully lit by the Sun and thus visible from the Earth's surface, proving once and for all who was capable of kicking the most butt.
They weren't the only ones considering this, apparently. In early November, 1957, the press in the U.S. reported a story that the Soviet Union was going to launch a nuclear weapon at the Moon, to coincide the explosion with the lunar eclipse on November 7th, thus commemorating the 40th anniversary of the October Revolution. Whether there actually was any discussion of such a plan by the Soviets is unknown, but since there was no such explosion on the Moon on that night, we can at least say that, if they planned it, they didn't succeed.
We can ultimately thank the late, great Carl Sagan for our knowledge of Project A119. He was the graduate student that did the calculations for the U.S. Military to figure out the potential effects of detonating a nuclear weapon on the Moon. Writer Keay Davidson discovered the information leading to the project's records while researching a biography of Sagan in the mid-'90s.
From the records of Project A119, it was apparently cancelled — in early 1959 — due to fear of public reaction, and the potential for disaster should the missile miss its target on the Moon and thus return to nuke some incredibly unfortunate area on Earth.
Those are certainly fine reasons to cancel a project like that, but personally I would hope that the unofficial reason was something more along the lines of "Come on now, this is just getting stupid, right?"
Given the crazy levels that the arms race reached, did anyone seriously think that kind of display would have had any effect other than to escalate things even further? We might now be looking at a vastly changed lunar surface, or be landing nukes on Mars instead of nuclear powered robot rovers.