Nanotechnology
Let's be clear here: nanotechnology is nothing more than a means to an end. Programming some sort of self-replicating von Neumann machine to eat the entire Earth up has its own massive problems (like, won't the ones at the bottom be crushed into their constituent atoms?), but even if it worked - you haven't destroyed the Earth. You've just got a planet made of nanobots that still needs destroying somehow. Program them to hurl themselves into space? Well, that's Meticulous Deconstruction, above.
Chilled
You will need: The capability to reduce the entire planet Earth to the microscopic temperatures necessary to cause it to revert to a Bose-Einstein condensate.
Method: It's well known and reasonably well-understood that substances at extremely low temperatures can get to the point where quantum phenomena start to have macroscopic, i.e. visible, effects. For example, it can just climb right out of a container, defying gravity. As to why, you would need some quantum physics under your belt.
Could the same work for a whole planet? Could a sufficiently cold body (if it were shielded from the heat of the Sun and ambient background microwave radiation) just spontaneously begin to dissipate into space?
Another idea is to use strong magnetic fields on the condensate to cause it to display what is currently referred as an unusual characteristic, undergoing something approximating a stellar supernova on a tiny scale: imploding on itself and then exploding, with a substantial fraction of the atoms involved disappearing entirely!
Feasibility rating: 4/10. The first idea may work, but the second one probably won't. This is because the experiment specifically used rubidium-85 atoms having a "negative atom-atom scattering length". I don't know what that is, but it sounds unusual for an atom, and we know for a fact that most of Earth is not made up of rubidium-85. Plus, the "disappeared" atoms didn't actually vanish, they just escaped the experiment system under high enough energy that they weren't detected escaping. And of course, generalising quantum phenomena to gigantic scales is never a great idea.
· Gamma Ray Burst'd
You will need: a star in Earth's stellar neighbourhood with >40 solar masses. Such massive stars are hard to come by; even Betelgeuse has only 20 solar masses. The best candidate I know of is Eta Carinae, which has over 120 solar masses but is ~7500 light years away.
Method: Gamma ray bursts are powerful, short-lived floods of gamma ray photons. GRBs come in two flavours, short (less than 2 seconds) and long (2 seconds to about 3 minutes); the latter are believed to be caused by stellar explosions called hypernovae, hundreds of times more violent than ordinary supernovae. Such stars are usually billions of light years away when they explode - the fact that we can detect them at this range should tell you enough about how powerful a hypernova is. So how about triggering one locally? Any such explosion within about 20 light years would probably be violent enough to destroy the Earth itself.
Feasibility rating: 0/10. This method was originally listed above, but astronomer Stephen Thorsett set me straight. It wouldn't work. Even in the titanic quantities described above, gamma rays wouldn't make a dent in Earth's actual, physical structure.
Sources: Lycurgus suggested this method.
· Burned away by muon-catalyzed fusion of the oceans
You will need: a supply of muons.
Method: The theory runs like this. A muon is a negatively-charged particle somewhat like an electron. If you dump a load of muons into some hydrogen, then some of the muons will replace the electrons in the hydrogen atoms. Because of the mass difference, the hydrogen atoms will suddenly get much smaller, causing the hydrogen molecules to be much closer together; enough that the probability of the hydrogen nuclei just randomly fusing with each other is high.
So, if you instead poured your muons into the oceans, they could cause the deuterium chemically combined with the water in the oceans to spontaneously begin undergoing fusion reactions. In theory, the amount of heat/energy released by the fusion of all the water in the world would be enough to destroy it by a good few orders of magnitude.
Feasibility rating: 0/10. All known muons decay in a few microseconds - fairly long for an exotic subatomic particle, but still too short to be practical, so unless you can generate your muons in bulk, for free, you don't reach energy break-even, and the fusion stops as soon as it starts instead of being self-sustaining.
Sources: Muon-catalyzed fusion was theorized in the late 1940s by Andrei Sakharov, and brought to my attention by Jef Poskanzer.
Comments: This method was never listed as plausible, but I put it up here anyway because the idea itself is intriguing, even if it wouldn't work.
· Blown up by vacuum energy detonation
You will need: some means of extracting huge amounts of energy from the vacuum.
Method: Some scientific theories tell us that what we may see as vacuum is only vacuum on average, and actually thriving with vast amounts of particles and antiparticles constantly appearing and then annihilating each other. It also suggests that the volume of space enclosed by a light bulb contains enough vacuum energy to boil every ocean in the world. Therefore, vacuum energy could prove to be the most abundant energy source of any kind. Which is where you come in. All you need to do is figure out how to extract this energy and harness it in some kind of power plant - this can easily be done without arousing too much suspicion - then surreptitiously allow the reaction to run out of control. The resulting release of energy would easily be enough to annihilate all of planet Earth and probably the Sun too.
Earth's final resting place: a rapidly expanding cloud of particles of varying size.
Feasibility rating: 0/10. This method was originally listed as plausible, but Alan Thomas set me straight: there are about five different ways to calculate the energy of the vacuum, all giving different answers. The methods which give the answers "large" or "infinite" are predicated on dodgy mathematics and almost certainly wrong.
Source: 3001: The Final Odyssey by Arthur C. Clarke
Allowing George W. Bush to continue to exercise his will on the world.
If you think this, you're completely missing the point. The power to destroy the Earth does not currently exist, and Bush's administration is not actively seeking to create such technology. Whatever Bush does, whatever the backlash from his policies on Iraq and oil and global warming, he cannot destroy the planet.
Paradoxes as described in Back To The Future Part II.
B
y definition, a paradox cannot actually come into existence.
Ceasing all thought
(if the Earth is not observed, then how can it exist?). Philip K. Dick said it best: "Reality is that which, when you stop believing in it, doesn't go away."
Semantics
A few people suggested exploiting a loophole in my mission statement and moving the Earth into orbit around a gas giant, making it a moon rather than a planet, or hurling it into interstellar space where it would become a wandering interstellar object. Yeah, yeah, very clever. Get back to work.
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Adding enough material to the planet Earth to cause it to undergo gravitational collapse and become a star instead of a planet
.The main problem I have with this is that the Earth is made mainly from heavy, pre-fused elements. Most of it is iron which simply won't 
undergo fusion at all. The amount of material you'd have to add to the Earth would be massive enough to be a star in its own right, and at the end of X billion years when it stops shining you'd still have a core of iron remaining in orbit around the Sun!
Of course, someone suggested you could add still more material until it becomes a star heavy enough to go supernova, so I'm going to come clean here: I have an irrational dislike of this method. It's not going in. Sorry.
· Detonating all the nuclear weapons ever created simultaneously, either all at one location or strategically placed around the globe.
This will irradiate pretty much the entire globe and kill an awful lot of people, animals and plants, but will actually destroy very little of the planet itself.
Proving that 1=0.
If 1 did indeed equal 0, so it is reasoned, then since there is 1 Earth, there must be 0 Earths... so, if one could prove it, the Earth would 
cease to exist. This is specious logic. Finding a proof in mathematics does not magically change a fact from being false to being true. It merely verifies rigorously as true a fact that always was true. Thus, if 1=0 could be proved, then it would always have been true and the Earth should never have existed. But Earth is still here. QED.
In fact it would be impossible for there to even exist a universe in which 1 was equal to 0. For any mathematical system in which 1=0, it is extremely trivial to prove, in addition, that 1=2, 2=3, and in fact that every number is equal. Or, in other words, the mathematical system has only one number in it, 0. In a universe which obeyed such laws, there would be nothing at all.
· Runaway fission at the Earth's core, as proposed by Tom Chalko.
It is true that while the Earth is mainly iron, there are significant quantities of other trace elements present, including fissile materials like uranium, thorium and - get this - radioactive potassium which have sunk to the core where latest studies suggest where they are indeed undergoing fission, generating heat and keeping the interior of the Earth warm. However, if a nuclear explosion did occur at the core, it would be insulated from the surface by sixty-three hundred kilometres of liquid iron.
· Gay marriage
I really have no idea why it is in here.
The final Curious @ phrase:
“The principle of equality does not destroy the imagination, but lowers its flight to the level of the earth”
(Alexis de Tocqueville)
