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Aliens featured Bill Paxton as Hudson, the wisecracking PFC. In the extended edition, he even claims himself to be the "Ultimate Badass". That said, it's kind of understandable that he would freak out (Game over man! Game over!) the moment the aliens actually showed up, since they are pure horror. However, despite varying between making sarcastic comments and making scared sarcastic comments, he actually proves useful throughout the film, and his "Last Stand" certainly makes him CMOA worthy.
He underscores his nature in his introductory song, and first line, "Stand aside, I take large steps!"
"Brave" Sir Robin in Monty Python and the Holy Grail, who has a troupe of troubadours to follow him about and sing of his praises. Unfortunately he runs away at the first sign of danger, and they incorporate his cowardice into their song...
Britt, the titular hero of The Green Hornet movie is pretty close to this throughout the film, boasting about how awesome he is, taking credit for his partner Kato's achievements and generally being a barely competent blowhard. There's even one scene where a villain is about to hit them with a cement truck. Kato picks up a rock to shatter the truck's front window so Britt can shoot his gas gun at the driver, but all we see of Britt is him running away, firing his gas gun in all directions shouting "EVERY MAN FOR HIMSELF!!!!" By the end of the film, Britt does gain a measure of competence.
John Kreese, the thuggish martial arts instructor from the first three Karate Kid movies is this and a bully of the worst kind. He talks like a Drill Sergeant Nasty among his students, encourages them to fight dirty, and even sunk so low as to order one of them to use an blatantly illegal move on Daniel, hoping to injure him to knock him out of the tournament. But in the beginning of the second movie, it becomes obvious he's hopeless in any kind of actual fight. When Miyagi stops him from strangling Johnny, Kreese tries to punch the old man, only for Miyagi to simply move aside, resulting in Kreese putting his arm through a glass car window; and if that weren't enough, he falls for it a second time, hurting his other arm. Even worse, for him, Miyagi then humiliates him in front of his students, comically tweaking his nose and letting him go when it looks like he's about to really hit him. (Showing himself to be the superior teacher both in skill and morals.)
In The Wild Hunt, Bjorn is the big cheese of his Viking faction of LARPers and is always talking in Large Ham voice about being a warrior and going on adventures. When shit gets dangerous, however, he completely freezes. A girl lampshades it by screaming at him to justify all his talk and do something.
Spence in Ronin, played by Sean Bean, makes himself out to be just as much of a deadly Bad Ass as all the other operatives chasing after the MacGuffin. However, he is eventually exposed as a poser and told to get lost, which he does.
English Bob from Unforgiven. Hero of countless DimeNovels and known as the Duke of Death. However, in spite of his excellent gun skills, he has no stomach for stand-up fighting, preferring to attack drunk opponents from behind.
Madmartigan from Willow is a subversion. He spends about half the film bragging about being "the greatest swordsman ever!" and the second half proving it.
Captain Harris is definitely this in the Police Academy movies. Ironically, while he chides the more likable cops because they're incompetent, he's no better, and a coward on top of it.
Curiously, although this Trope is the Inverse of Cowardly Lion, the Trope Namer for that one fit this Trope when he first appeared in The Wizard of Oz, daring both the Scarecrow and Tin Man to fight him, saying he take them both at once (with one hand behind his back, standing on one leg, and with his eyes closed). He lost the attitude quickly.