It is not only possible, but the easiest option, to "blow them up Armageddon style" (minus the drilling and the like). There's a lot of simulation work going on right now and the results have been consistently encouraging that even a small nuclear weapon could obliterate quite a large asteroid into little fragments that won't re-coalesce, while simultaneously kicking them out of their current orbit. A few years ago they were just doing 2d calcs, now they've gotten full 3d runs.
Think for a second about what nuclear weapons can do on Earth. Here's the crater of a 100kt nuclear weapon test. It's 100 meters deep and 320 meters wide. You could nearly fit a sizeable asteroid like Itokawa inside the hole. And that thing had Earth's intense gravity field working against it and was only 1/10th the size of weapons being considered here. In space you don't need to "blast out" debris with great force like on Earth, you merely need to give it a fractional meter-per-second kick and it's no longer gravitationally bound. And the ability of a nuclear shockwave to shatter rock is almost unthinkably powerful - just ignoring that many if not most asteroids are rubble piles and thus come already pre-shattered. Look at the "rubble chimneys" kicked up by even small nuclear blasts several kilometers underground (in rock compressed by Earth's gravity). Or the size of the underground cavity created by the wimpy 3kT Gnome blast - 28000 cubic meters. Just ignoring that it had to do that, again, working against Earth's compression deep underground, if you scale that up to a 1MT warhead the cavity would be the size of Itokawa itself.
You of course don't have to destroy an asteroid if you don't want to - nuclear weapons can also gently kick them off their path. Again, you're depositing energy in the form of X-rays into the surface of the asteroid on one side. If it's a tremendous amount of energy, you create a powerful shattering shockwave moving throughout the body of the asteroid. If it's lesser, however, you're simply creating a broad planar gas/plasma/dust jet across the asteroid, turning that whole side into one gigantic thruster that will keep pushing and kicking off matter until it cools down.
The last detail is that nuclear weapons are just so simple of a solution. There's no elaborate spacecraft design and testing program needed - you have an already extant, already-built device which is designed to endure launch G-forces / vibrations and tolerate the vacuum of space, and you simply need to get it "near" your target - the sort of navigation that pretty much every space mission we've launched in the past several decades has managed. In terms of mission design simplicity, pretty much nothing except kinetic impactors (which are far less powerful) comes close, and even then it's a tossup. Assuming roughly linear scaling with the simulations done thusfar, with enough advance warning, even a Chicxulub-scale impactor could be deflected / destroyed with a Tsar Bomba-sized device with a uranium tamper. Even though it was not designed to be light for space operations, its 27-tonne weight could be launched to LEO by a single Delta-IV Heavy and hauled off to intercept by a second launch vehicle.
