There is much modern technology that was inspired by the original Star Trek - for instance, flip-phones were designed to look as much like Kirk's communicator as possible. (Kirk's communicator, though, couldn't access computer networks, display images, or play games.)
Transporters seem to be even theoretically impossible - so far.
Warp drive as depicted may or may not be possible, as it relied on an understanding of physics far beyond our own (knowledge of the existence of subspace, realms of existence in a superspace, not dissimilar from modern brane theory). However, Miguel Alcubierre did come up with a theoretical way to exceed lightspeed without violating Einsteinian physics, based on the idea that while an object cannot go faster than light, the expansion of space itself can. His theory would compress the space in front of a bubble of space containing the craft, and expand the space behind it - the ship would remain in flat space, but the bubble would be riding the compression and expansion of space. The problem with Alcubierre's theory was that it required a mass of exotic matter about as great as that of Jupiter. However, in recent years, Howard White worked out a modified version in which the mass would be a flattened toroid, and the edges of the warp field would be vibrated; this would give effective "velocities" of up to ten times the speed of light, and reduce the exotic-matter requirement to about 500 kilograms or so. Of course, thus far we have no idea how to produce exotic matter, and can't even prove it exists; hope is given by discovery of a Higgs boson, but may be dashed if this Higgs boson violates the standard model (a matter still in question). However, NASA is constructing a testbed on which to examine Alcubierre/White warp theory at a microscopic scale. If they can produce tiny little warp bubbles, it will prove the theory correct - then we can work on how to get exotic matter.
The rest of the technology is at least possible. For instance, a researcher in California, working under a military contract for non-lethal weapons, has developed a phaser - it fires a low-power laser to ionize the air between the emitter and the target, then a phased electrical pulse to follow the ionized channel to the target. The problem with the device currently is that it's too big to be carried in anything smaller than the bed of a pickup. However, given how technology has shrunk the size of devices over the years, I wouldn't bet against a handheld version within 200 years...
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Sodium is a metal that reacts explosively when exposed to water. Chlorine is a gas that'll kill you dead in moments. Together they make my fries taste good.