The X-Files/Fridge

Fridge Brilliance

 * An all-powerful evil conspiracy... could not oust rogue Agent Mulder and sympathetic Assistant Director Skinner from the FBI, or at least remove them to some insignificant bureaucratic duty? Explained (rather poorly) by the back story of Mulder having been regarded as a hero who aided in the capture of a vicious serial killer early in his career, and therefore any action against him would be a public relations pain in the backside. Not explained by the fact the Evil Conspiracy had already ousted people far more important and powerful.
 * This Troper always thought it was because no one liked or believed Mulder; this way, anything he actually found out was guaranteed to be treated as a Cassandra Truth. In a way, Mulder was an extra layer of protection.
 * The conspirators may also have found him a convenient stalking horse, when investigating strange events that they weren't involved in. All those Monsters Of The Week, for example.
 * Take it even further, actually: the Cigarette-Smoking Man always paid close attention to wherever and whatever Mulder went and did, and there are several hints throughout the series that, when it didn't interfere with their specific projects, the "cabal" had no problems implementing whatever safety procedure suggestions he would make regarding dangerous unexplained phenomena. To put it another way, Mulder was half a whistle-blower, half SCP Foundation agent. They would've been fools to have actually gotten rid of him completely, and they knew it.
 * It was likely a useless move. Mulder has shown that it doesn't matter if he is in the FBI or not--he'll go after the truth with or without their help.

Fridge Logic

 * Quite a bit, but the most notorious example might be from Fight the Future: how did Mulder and Scully get back to civilization from the crashed spaceship in the Antarctic? (There's a joke about this later in the TV series.)
 * Darkness Falls episode (1994) does it at least twice. The Monsters Of The Week, basically vicious blood-sucking bugs, are "afraid" of light in Mulder's words, in practice they are inactivated by light. Our heroes and a few others are caught at night in stranded cars, but with fully operational engines and electrical systems. Why didn't they turn on the headlights and interior dome lights? And how could the bugs after killing a 70-90 kg (155-200 lbs) man raise his body onto a tree? (Although the last one can be explained by the fact the victim could have been killed while he was already up in the tree, since he was a logger).
 * There was supposed to be critically short fuel supplies, making escape (or even cabin lighting) via the vehicles impossible. What struck this troper as odd is why they didn't maintain a small to medium reserve to use as an extreme form of lighter-fluid, and use it to start campfires that, once going, would have the unlimited fuel of the forest.
 * Speaking as someone who's grown up in The Other Rainforest, at that location and time of year, any wood they didn't have chopped and stacked under cover was probably too damp to burn. Even lighter fluid sometimes isn't enough to keep a fire going, if it's too soggy.

Fridge Horror

 * In "The Calusari", we know Michael, who is can travel around and physically harm people. He's a lot creepier and more emotionless than Charlie...which is exactly how the boy in the opening appears. Which means that kid might have been Michael, not Charlie, which then makes one wonder, how often has that happened? How many times have the parents walked around thinking they were with Charlie, and instead had ?
 * In the first part of the Season Eight finale, Doggett calls his friend Agent Crane in on the weekend to help investigate a murder, which Crane mentions caused him to miss his "kid's Little League game." Then at the end of the episode we find out that Crane is one of the alien Replicants with the bumpy metallic spines. Assuming that he does have a family, they're living with a murderous alien and are absolutely none the wiser...