Siding Spring now passes about 10 Earth diameters away from Mars.
What are the estimates of how often comets fly by Earth at different distances? Are the distances to some famous historical comets known?
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The 1910 pass of Earth was especially close and, thanks to expansive newspaper coverage, eagerly anticipated by the general public. In fact, Earth’s orbit carried it through the end of the comet’s 24-million-mile-long tail for six hours on May 19, earning the story the day’s banner headline in The New York Times. While most reporters of the day turned to astronomers to get the facts straight, the yellow press chose to pursue the story in more fanciful ways, helping to fuel the fears of the impressionable that the end of the world was nigh.
Halley's got within 5.1 million kilometers of earth in 837 CE
On the Near Earth Object Program web page from NASA, it lists about 80 objects that passes near the Earth over the next 4 months. Many of them pass within 0.01 AU, the closest one will pass on Oct 24 and be only 0.0038 AU (or 1.5 times the Earth Moon distance) away.
Scientists estimate that several dozen asteroids in the 6 to 12 meter size range fly by Earth at a distance closer than the moon every year ...
It is also predicted that in April 2029, 99942 Apophis will pass by the Earth at a distance of 38,300 km (inside the orbit of geostationary satellites).
The answer is that they don't come very close. As Wikipedia notes,
It is rare for a comet to pass within 0.1 AU (15,000,000 km; 9,300,000 mi) of Earth.
Even better, though, is an actual list of some of the closest approaches of comets to Earth. The closest one listed, Comet Lexell in 1770, came 0.0151 AU away from Earth. The list only shows 20 comets have come with 0.1 AU of Earth. So Earth has not had a close encounter with a comet (like Mars is having) in a long time, if ever.
This site says that C/1491 came within an incredible 0.0094 AU in 1491, but I'll take the IAU site over it. And it does admit that that comet has an uncertain orbit, so that figure could be wrong.
As for the distances to famous comets:
Also, some near-Earth asteroids used to be comets. Extinct comets have lost a lot (if not all) of their ice and so may be classified as asteroids. This blurs the lines between the two groups. Some extinct comets have become near-Earth asteroids. This explores the issue pretty well. So if you include extinct comets as "comets," you may have comets that have come even closer to Earth.