There was just a show on either Discovery or Nat Geo where the world's top scientists talked about that very subject. Turns out the Earth at least is very sturdy. The ways they came up with that would actually work are:
1. Create or coax a black hole with the same mass as the Earth (which would be the size of a marble) into orbit around the planet. it would stretch the planet apart and then absorb it.
2. Dump a trillion tons of antimatter into the Earth's core.
3. Coax a series of large asteroids into a conveyor belt orbit between Jupiter and the Earth. With each pass, some of Earth's momentum would be stolen, and after 1000's of years, the Earth would fall into the sun.
4. Asteroid collision. You would need something with around 1/2 Earth's mass to destroy it completely. A 60KM object or less would make Earth uninhabitable.
5. Self replicating nanobots. Nanobots that can reproduce themselves out of individual atoms, and then each generation helps make the next generation is probably the most feasible situation. The Earth would be consumed and turned into a blob of robots.