There are a number of what appear to be obvious flaws in the "science" of Star Trek, and an equal number of explanations that attempt to explain those flaws as misunderstandings.
Sound and light in spaceEdit
A feature of almost every episode (as well as most non-Star Trek science fiction television shows and movies) is the reverberations of sound: the Enterprise blasting into warp, firing the phasers, villains' ships exploding. But sound is the vibration of a medium. Space is a vacuum, and therefore devoid of matter, including any medium (e.g., air) for sound to travel through, so no sound is ever possible.
Response: Much of the noise in Star Trek involves things that can be heard from within the ship. For example, when a phaser is fired or when a ship is being hit by a phaser, presumably there is a noise that can be heard within the ship. It may also be noted that when a ship or other object in space explodes, it explodes into (usually small) pieces. When these remaining particles come into contact with another ship, including the one that fired the destroying torpedo or phaser beam, they set up vibrations within that ship which are generally called "sound". The primary issue with "hearing explosions in space" is not that they are heard, but that they are heard too soon and would likely sound more like thunder. Additionally, some of these sounds could be rationalized as acoustical feedback the sensor systems provide for the ship's crew. Some present-day radar systems, especially in the military area, already do this. Another explanation is that the sounds in space have been added simply for dramatic effect; a battle without sound might have been perceived by producers as boring. Like many action-oriented SF shows, Star Trek features battles between spacecraft, many of which use some sort of energy weapon. Early in the original series, the Enterprise crew members used lasers (see, e.g., the pilot episode "The Cage"). These "lasers" were either red or blue depending on the era. Unfortunately, science dictates that without a reflective medium, photons are not scattered out of their direction of travel. A laser beam passing through space can be seen only if dust particles present in space scatter its light or if it heats up the gas present in space enough to create a trail of glowing plasma. Simply put, a laser is invisible in a vacuum.
Response: References to lasers are incorrect and should be retconned out, especially considering that Star Trek: Enterprise uses "Phase cannons" and "Phase pistols" (although this itself contradicts a line in Star Trek: The Next Generation that phasers didn't exist in the 22nd century, though this is assuming that Phasers and Phase Pistols are the same technology). The beams in Star Trek are phaser beams, and as such, they emit light when going through a vacuum. Sources like the Star Trek Technical Manual indicate that phaser weapons emit beams of "nadions", one of many fictional particles invented for TNG. Presumably, a nadion beam can emit light even in a vacuum; nadions might spontaneously decay into photons, for example. There are no physical laws forbidding this process, since nadions have whatever properties the Star Trek writers give them. Also, like with sound, a battle involving only invisible weapons would be incredibly boring and confusing for viewers.