Why accelerating to light speed and beyond is impossible, is because particles with a rest mass cannot reach the speed of light. Likewise, mass-less particles such as photons can't exceed the speed of light, and only particles with a mathamatically 'imaginary' mass can travel faster than light- but only faster than light.
The argument against other forms of effective FTL (i.e. you do not locally exceed the speed of light, but you arrive at a destination before a photon emitted at your starting point), is that they are either physically impossible (ruling out one method) or that they create causality violations (time travel) and thus are impossible in physics (ruling out any for of effective FTL). It has, however, been proposed that either quantum effects could disallow time travel via effective FTL, or that time travel is possible and systems are at work to prevent paradoxes- i.e. the many worlds hypothesis (paradox spawns a new timeline) or the Novikov self consistency principle (the probability of a paradox occuring is 0). This, like everything else in this discussion, is highly speculative.
The problem with the Alcubierre drive is not only that you have either problems with needing tachyonic matter or generating a naked singularity outside the bubble while travelling FTL, but also that the bubble has steering problems and that if you cross the 'light speed barrier' some sort of horizon thingy forms from all the atoms that you crash into, which heats the interior of the bubble up to a point where everything is quark-gluon plasma. That and the fact that the bubble needs not only an incredible amount of mass, but that also that some of this mass needs to be negative matter, which is purely hypothetical. Subsequent papers have shown ways to decrease the amount of mass needed, but such a project would probably still be monumental in scale.
In short, if you have negative matter, and you can create such a 'bubble' (another issue), you can potentially use this method for propulsion, but only at velocities lower than c... which makes it far less interesting and inexorably less useful.
The problem with wormholes is again, that they require negative matter (to prop open the "throat" of the wormhole), and they need to be very massive (on the order of planetary masses, at least). To travel through a wormhole, you need one of the 'mouths' of the wormhole to already be at the destination, and it has to be shipped their physically, at a velocity lower than c. This means that I cannot, for example, make a 'short dash' to Alpha Centauri.
Shipping a wormhole at relativistic speeds also means that you will travel back in time when you travel to the destination wormhole. A wormhole network can be constructed to prevent paradoxes, but it would still be a little unnerving to leave Earth in 2011 and arrive at Gliese 581 in 2008...
I can't say I know much about FTL in Heim theory, although it does look very interesting, the paper was not peer reviewed, does not seem to be well understood, and exists on the fringes of science.
. You leave Earth orbit and keep on accelerating. After a while you get close to light speed. Your ship starts to get heavier, because of the theory of relativity. Now you know that the heavier something is (more correct the more mass it has, of course) the harder it is to accelerate. You can get reaaallly close to the speed of light, but you can't reach it.
