The Double-Slit Scanner
The double-slit experiment shows that particles travel in ways that don't always agree with the predictions of classical physics. I have heard some its findings described as essentially proving that some particles simultaneously travel (or evaluate) multiple paths through the room (which would require the particle to travel many times the speed of light, since it would need to complete all the exploratory routes as well as the route that observers actually see). As crazy as that sounds, it seemed to be the only way to explain the facts.
So let's extend my poor layman's version of this to the cosmic scale.
I'm going to assume that any kind of warp bubble will violently displace all the particles in the immediate area of the vessel, and that displacement follows the vessel as it travels. Space is not completely empty, after all. Within any star system, there will be the heliosphere: a colossal soup of charged particles that engulfs the entire solar system, flowing outward from the star. To me, that giant, electrically charged structure seems a lot like a spiderweb (which is, after all, Mother Nature's radar).
This displacement can be detected remotely, faster than light waves would propagate, by observing the movement of subatomic particles at a monitoring site, because those particles are affected by the warp bubble's displacement in much the same way that, in the double-slit experiment, changing one slit mysteriously has an effect on the particles that "only" go through the other slit.
Now, that might not be enough to detect a vessel on the other side of the solar system that activates its warp bubble for only a very short time. But if it travels at speed for tens of minutes, the bubble will carve a giant tube through the heliopause, and I imagine that would be adequate to both detect that FTL is happening and locate roughly where it is.
You'll have to decide whether it's practical for this detection apparatus to be mounted on a vessel rather than being on the ground or a space station. However, I would suppose that the detector would be useless while it is inside a warp bubble, i.e. while the vessel its on is at warp.
The heliosphere only covers star systems. There is, of course, dust and gas in the void between the stars, but it might be too dispersed to act as a spider web. If so, that would mean that vessels traveling at FTL can only be detected once they enter a system, and that a vessel operating in ultra-deep-space could not be detected beyond lightspeed range.
I would also expect that this device would not be able to tell you anything at all about the stuff inside the warp bubble. All you would know is that a warp bubble is moving along a particular trajectory: the inside could have any kind of ship, or an alien, or a freak asteroid that creates a warp-ish bubble around itself.
The DSS also makes margaritas.