Anthropomorphic Personification: "Guilt Trip", in which the personification of Guilt is forced to go on vacation for messing up on the job, and meets and falls in love with the personification of Love.
Beauty Contest: In "Miss Stardust", one is crashed by an alien ("Weird Al" Yankovic) who threatens to destroy Earth unless the contest lives up to its name — Miss Galaxy — and allows contestants from other planets to participate (and ideally win). To make matters trickier, the new entrants aren't Human Aliens...
Losing Your Head: "Go to the Head of the Class" has Sadist Teacher B.O. Beanes, after accidentally being killed, coming back to life with his head separate from his body because the picture used in the resurrection spell got torn in two.
Monster Is a Mommy: Inverted/parodied in "Mummy, Daddy", where an actor trapped in a highly constricting mummy costume frantically attempts to reach the hospital where his wife is giving birth. His task is further complicated by two things: a hostile band of backwoodsmen and a real mummy.
Mood Whiplash: As each episode is done very differently this happens often and can catch people off guard. One episode will be whimsical and comedic, then the next will be dark and serious.
Not-So-Phony Psychic: "The Amazing Falsworth" is a play on this, where people believe he's an act but he really can read minds.
Oh Crap: "The Amazing Falsworth" is one big episode of this. Falsworth does his mind-reading act and winds up picking up on the thoughts of a serial killer. Worse, he was blind-folded at the time, so he doesn't know who the killer is.
Spinoff: Family Dog, although picked up by CBS instead of NBC. The series was notably done without the input of Brad Bird, who intended the original episode to be a stand-alone outing and didn't feel the concept could support a series. (Given its Troubled Production and the fact that CBS cancelled it after five of the ten produced episodes aired, he was right.)
Stage Magician: The main subjects of "Mr. Magic" and "The Amazing Falsworth".
Strange Minds Think Alike: In "Family Dog", Ms. Lestrange promises to turn the dog into "a quivering, snarling, white-hot ball of canine terror." Later, when the dog attacks burglars, one exclaims, "He's turned into a quivering, snarling, white-hot ball of canine terror!"
Why Won't You Die?: The characters in "One For The Road" keep trying to kill the Disposable Vagrant with lots of alcohol, kerosene, and walks in the freezing cold, but he just won't keel over. He never does.