For me the best argument against FTL travel is that, according to relativity, you have either FTL travel or causality, but you can't have both.
Imagine it this way. Let's assume we have a rocket, that travelled from the Earth to Alpha Centauri in a year (as perceived by people on Earth), achieving this way an average velocity of 4.2 c. We then have two events (in form (t,x) relative to Earth):
(0,0) - start of the rocket
(1,4.2) - arrival of the rocket on Alpha Centauri.
Now let's say we have a regular, slower than light rocket, which travels in the direction of Alpha Centauri at 0.5 c, starting in the same place and time as the previous rocket. Let's calculate what will people on that rocket see...
Using the Lorentz transformation, we get that the first rocket still started at (0,0). But the coordinates of the arrival will be:
[math]x' = \frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}} = \frac{4.2-0.5}{\frac{\sqrt{3}}{2}} = \frac{7.4}{\sqrt{3}} \approx 4.27 ly[/math]
[math]t' = \frac{t-\frac{v}{c^2}x}{\sqrt{1-\frac{v^2}{c^2}}} = \frac{1-2.1}{\frac{\sqrt{3}}{2}} = \frac{-2.2}{\sqrt{3}} \approx -1.27 yr[/math]
Wait, what? t' is negative, which means that relative to the second rocket, the first one arrived 1.3 years before it took off! Let's say that the first rocket was carrying a message about a great war that started on Earth just before its takeoff. According to the second rocket, people living in the space station near Alpha Centauri were notified about the war before it started. Causality is broken.
The point is, if the interval between two events is spacelike ([math]c^2 (\Delta t)^2 - (\Delta x)^2 < 0[/math]), there always exists such a frame of reference, that the sequence of events is reversed in that frame. This means that if you can travel faster than light, you can make a causal connection between a sequence of events, which will be reversed in another frame of reference. Since it is widely believed that causality should work, this means that there can be no FTL travel.
