Even Batman has had this problem. In the Batman/Superman crossover episode, Batman gets his cape caught in the Daily Planet's printing press while fighting a robot drone. The cape and cowl are pulled off, revealing his secret identity to Lois Lane.
In the typical DC universe, Batman's cape clasp has a quick-release mechanism that automatically detaches his cape from his cowl if it gets caught in something specifically for this reason. Of course he'd have a detachable cape, this is Batman we're talking about! Anything a crook might try to use against him, he's already thought of and planned for. For all of this, it helps that the cape has (in many versions) a practical function: as a glider, or at least a parachute (since Bats is of course a Badass Normal with no actual superpowers).
In Batman: Gothic, it's even noted, that in addition to a quick release function on the cape, the part where it attaches to the cowl also has a built in neck brace to protect him from whiplash or getting his neck broken if the cape is caught by something especially heavy or fast.
Watchmen: In the Backstory, the corporate hero Dollar Bill was ordered by his sponsors to get a cape as a part of his outfit, in order to increase his marketability. However, one day, when he tried to stop a bank robbery, his cape got caught in a revolving door, allowing the bank robbers to shoot him to death. Nite-Owl notes (somewhat bitterly) in his memoir that Dollar Bill would likely be alive today were he allowed to design the costume himself.
The page image is from a time when Captain America briefly operated as the superhero Nomad.
Subverted in All Fall Down. Wearing a cape is what saves Paradigm's life.
The Scarlet Witch had a pair of cases in Avengers West Coast, and tried a new costume without cape. It didn't stick.
Given an Ironic Echo of sorts when Syndrome's cape gets caught in a jet turbine, leading to his death.
In Mulan, Shan Yu is defeated when Mulan stabs his cape with his sword holding him in place so Mushu can shoot him with a rocket.
In the Golden Films version of Aladdin the villain Hasseem during a sword fight with Aladdin accidentally trips over his cape and ends up stabbing himself.
Films — Live-Action
Subverted in The Dark Knight during the Batpod chase. The costume makers designed Batman's cape to fold into a backpack using the same memory cloth tech that makes the glider, as they thought it would be snagged on the wheel. However, during test runs the cape flowed freely without getting caught, which looks fucking awesome. Applied for regular fist fights however.
Inverted in Superman Returns. Superman is saved from drowning because Lois is able to grab him by the cape.
In Soon I Will Be Invincible, supervillain Dr. Impossible admits that his cape gets in the way during the inevitable fight scene, and is uncomfortable in hot weather, but feels that the psychological advantage imparted by a dramatically billowing cape outweighs the disadvantages.
Subverted in, of all places, The Greatest American Hero — Ralph's cape only extends down to his rear end and is just about the only part of his Supersuit not to give him problems.
Calvin and Hobbes: In one of Calvin's earliest attempts at playing a superhero, strikes a dramatic pose in his cape and says, "This is a job for..." He then proceeds to trip over his cape and fall to the ground, before finishing his sentence: "...someone else."
The pro wrestler Ultimo Dragon usually wore a cape to the ring. But at WrestleMania 20 he humiliatingly tripped over it during his entrance to the ring (which, if you watch WWE, is pretty much the worst possible time you could have an accident like that).
The inevitability of a Cape Snag happening in real action is precisely why every wrestler who wears a cape takes it off before entering the ring.
Evil Inc. researched "the cape gun", which is fired at the hero. If it hits the cape, it spins and winds up the cape, strangling the hero (or the intern they were demonstrating it on).
Homestuck features a minor incident in which an ally of the caped hero gets caught in it while they were grappling over a mix of cabin fever and culturally incompatible conceptions of a reasonable romantic setup. The guy with the cape had to suplex the guy caught in it to free him.
A version of Batman's detachable clasp is shown in Batman: The Animated Series when his cape becomes trapped in the grip of a robot Joker henchman — Batman quickly detaches his cape and spends the rest of the episode capeless.
Batman's cape got snagged on a Daily Planet press in Superman: The Animated Series. Apparently it was attached to his cowl, and its removal exposed his identity to Lois Lane.
In Teenage Mutant Ninja Turtles, Michelangelo decides he wants to be a superhero and sets himself up with a costume, complete with cape... which proceeds to trap him in the door of a bus falling into the river. Oops.
In one episode of Darkwing Duck, the titular hero got his rip proof cape trapped in the hood of a car just as a giant monster was bearing down on him. Later in the episode, it was shown that Negaduck was Genre Savvy enough to buy rip-able capes.