In the Dennis Leary series “Rescue Me” about firefighters, the main character spends the first season or two being haunted by the ghosts of fire victims. They follow him around and speak with him. Then they all mysteriously went away after the first season and were never mentioned again.
I worked at the front desk of a hotel back in college, and we had a dedicated, internal phone for guests calling us from their rooms. One of the other guys would sometimes just pick up and answer the phone without it even ringing, and it seemed psychic to everyone else. Actually, the trick was that the ringer would make a very subtle click just before the bell started ringing. Once I learned to listen for the click, I could do it too, though I would usually let it ring once first. Wouldn’t explain how someone would know who is calling, though.
According to the book Radar got his nickname by being able to anticipate things that no one else could.
In the movie he became a hyper-competent clerk, completing things a half step before his commanders requested them.
I have blocked much memory of details of the television series.
Don Draper sees a number of ghosts over the course of Mad Men including wraiths of his brother, Anna Draper, Rachel Mentz and Bert Cooper. Some may be hallucinations, dreams, etc but “seeing” Mentz and then finding out the next day that she just died days before feels like more implied than just a daydream or nerve-ridden mind tricks.
But as Brian’s life paralleled Jesus’, would this be the parallel to the Devil taking Jesus to the desert to tempt him? “cast yourself down” etc. At least, that’s the interpretation I’ve always had.
That episode was subtle enough that I didn’t realize the Juliana Hatfield character was supposed to represent a ghost until Angela’s mom asked her, “How did you die?” I was like, “Wha…???”
Funnily enough, although as you say my country is altogether heathen, I’m totally not! Still get to have a sense of humour tho’ 
Well, if you can stand the challenge of “Life Of Brian” AND laugh about it, your faith must be really firm and I can only congratulate :D!
“Barney Miller” writers liked to occasionally introduce paranormal characters. It was an inspired counterpoint to the general realistic ambience of the 12th Precinct.
Off the top of my head, besides Dietrich’s admirer from the future, a character who may or may not have been the returned Jesus Christ made an appearance.
Usually they played it down the middle, i.e. kept the supernatural aspect ambiguous, so that in the end both the other characters (cast) and the audience would be left scratching their heads.
But there was also an episode in which Kenneth Tigar played a would-be werewolf. That time, they crossed a line and included an “Exorcist”-like scene in which the eerily-lit Tigar bounced around in a frenzy inside the famous holding cell, including hanging upside-down from the bars.
I’m currently re-watching the '90s Canadian/American buddy-cop/fish-out-of-water/adventure/comedy/drama Due South. The Mountie main character, Constable Benton Fraser, is a comically omni-competent super-cop, but within the context of the series, he’s just an unusually capable and exceptional guy. There are no supernatural elements at all. Except…
Starting early in the first season, Benton Fraser is “haunted” by the “ghost” of his late father. For the most part, the show tries for some ambiguity as to whether the father is a ghost or just a figment of Benton’s imagination, but it’s pretty clearly a theatrical conceit to portray the character’s inner dialogue, and there’s absolutely nothing to indicate that the father has any objective reality outside of his son’s mind. Except…
In one episode, involving the father’s murderer, the father’s “ghost” pretty clearly uses some sort of psychic ability to influence the thoughts and actions of a corrupt cop who’s trying to kill Benton and the father’s killer (it’s complicated). And then the father’s ghost clearly, physically, objectively, appears to his killer. These are abilities the father’s ghost never displays in any other episodes.
Also, one episode features a homeless man who has genuine clairvoyant, psychometric, and precognitive abilities. There’s absolutely no ambiguity. He’s a genuine psychic. And he and his abilities are never seen or mentioned again. (I’m guessing he won James Randi’s million dollar challenge and retired to a life of luxury :)).
The last episode of Due South deals with the fate of Benton’s father’s ghost beautifully.
Was this after season 2? The quality went way down after season 2.
Hence Frasier’s comment to Niles about her ‘abilities’ in Episode 3: “We’ve decided to find it charming”
But there were a few times where she successfully uses the shining.
Someone once lost a ring or something and Daphne was like, “It’s in the bathroom. No, it’s in the bedroom. No, it’s in the hallway,” and then Eddie comes scampering out of the hallway with the ring.
Another was Daphne having a vision of her true love with a dragon, then we see Niles receives a dragon statue as a gift.
Knots Landing, (does anybody remember Knots Landing?) had an episode where some of the women take refuge from a storm in an abandoned house where ghost children almost kidnap Val Ewing.
In terms of Daphne / Radar etc, I think with individual characters, it might be difficult drawing a line between genuinely implying the supernatural and dialling a character trait to 11 for comedic effect.
For instance, there are probably countless sitcom examples of a glutton character showing up and somehow knowing that, say, cake was going to be given out, in a situation where they had no possible way to know that.
Or letting them be mysteriously right about something just for the sake of a throwaway gag.
Movie like Die Hard
Where the Hero can be repeatedly kicked in the ribs by a really big guy wearing steel-tipped construction boots, later walk barefoot over glass leaving bloody tracks, yet at the end of the movie is well enough to make out with the girl.
Wot? You don’t think having regenerative/healing powers like that is supernatural?
Nope, both Season 2 episodes.
In the novel You Only Live Twice, James Bond prays to some stone idols for success in his mission to kill Blofeld, and both he and Kissy Suzuki see the idols nod and grant the prayer. Bond immediately denies it.
Regards,
Shodan
In the wonderful, long-forgotten ‘Days and Nights of Molly Dodd’, Molly goes to a psychic who makes a bunch of utterly ridiculous ‘predictions’. And adds, ‘oh, and some man is going to go after you, hon. Beware.’ … that part turned out frighteningly true. (I loved that show, it had an air of unreality, each episode dream-like, special.)