For some reason, the character is always standing at the intersection of Strange Street and Bizarre Boulevard. They run into situations or creatures that most people don't even believe in, much less have to deal with on a regular basis. Through no fault of their own, they constantly suffer through the effects of the paranormal and supernatural. Alternatively, the character may not think of the weirdness about them as particularly strange; after all, this sort of thing happens to them all the time. However, if something starts getting surreal on the show, chances are, they're at the center of it.
Often seen in comedy, especially when the writers get lazy and don't even bother to Hand Wave their plots anymore. Also used egregiously in Sci Fi and fantasy series, with the chain of weirdness catalyzing in the pilot, and each specific occurrence resolved at the end of the episode. Bonus points if, at some point, one of the characters brings it up and questions, "Why does this kind of stuff keep happening to me/us," or notes that that "Ever since [the events of the pilot happened], you've been a magnet for the freaky."