Lemming Cops

Police involved in a Hot Pursuit will not make even the slightest attempts to preserve their own safety or welfare, or that of Innocent Bystanders. Whatever risky or insane driving the bad (or sometimes good) guy does, the cops will simply follow in his tire tracks, trying to imitate everything he does, no matter how much injury to themselves or damage to their car will result.

If the bad guy plows though a farmers' market, the cops will do the same to chase him.

If he jumps an open drawbridge, the cops will do it too (or at least try).

And if one cop wrecks the rest will mindlessly plow into him, sometimes without even making an attempt to avoid a wreck, 'cause the resulting pile of twisted metal is cool.

Sometimes this is done for humor, but most action movies/TV shows portray this as normal police behavior.

When police are the designated Mooks of a story, they are often also very bad drivers who will probably wreck just driving normally, which only adds to this trope.

Combining this and the fact they have guns makes them an easy stand in for the Redshirt Army in situations where the military isn't likely to be involved and possibly also Adults Are Useless if the protagonist is a kid.

In Video Games, this tends to get Turned Up to Eleven with cops intentionally trying to crash into the player at full speed and ignoring everything else.

Of course this is quite different from Real Life where police have training and procedures for pursuits. Watch any TV program on the subject and you will see the police are very careful, to the point of backing off and following by air if it looks like civilians will be harmed. Real Life cops will generally terminate a pursuit when it becomes too fast or dangerous.

And that's all we'll say about it.

Anime and Manga

 * In Full Metal Panic!? Fumoffu, this is played for laughs when an obstinate police officer (who looks suspiciously like she wandered off the set of You're Under Arrest) gives chase to Sousuke and Chidori double-riding a stolen bike using her squad car and fails epically.
 * The next time we see her, she mentions she's on probation due to wrecking the car.
 * Panty and Stocking With Garterbelt. dear God, Panty and Stocking with Garterbelt. This has to be seen for yourself. Still, it may be a bit of a subversion in that the cops are on the protagonist's side, and they actually manage to do some real damage, even if it is, like expected of anything of lemming-like persuasion, the most pyrrhic victory ever.

Comics

 * Sin City has wave after wave of cop cars getting wrecked or blasted by hookers in many stories that feature car chases. Leading a squad car into Old Town is the best way to avoid the cops.

Films

 * Movie example: Smokey and the Bandit.
 * Parodied brilliantly in the Blues Brothers (picture above), with an unbelievably large number of lemmings cops. (Made funnier in that, since the heroes are driving an old police car, the cops should be able to do what they do.)
 * Don't forget that the Blues Brothers were on a Mission From God.
 * Taxi 2. Dozens of police cars follow the main characters. Wherever the main characters' car goes, the police go as well—all of them. The damage caused by the followed car is nothing compared to the destruction caused by the cops. Eventually one cop wrecks and all of the others plow into him and into the resulting wreckage. Not one police car survives. Accompanied by a great sound track, as well.
 * Batman Begins. Despite Batman's "no killing" creed (which the cops don't know about), many cops engage him in a high speed chase through highways and back streets. Several spectacular crashes occur with 2 or 3 car pileups, grenades are deployed at one point, and the Batmobile even drives over a police cruiser, demolishing it. When he gets back to the Batcave, Alfred (who has been following the news reports of the chase) tells him "It's a miracle no one was killed." Er, yeah.
 * They try to justify it in the direct sequel The Dark Knight, by giving the Batmobile "lifesign sensors." It knows which cars are okay to blow up.
 * Also, Batman Returns: Batman's coming up to a narrow alley, pursued by a few cop cars because the Batmobile was being controlled by the Penguin in an attempt to frame him. Batmobile jettisons stuff to become the Batmissile, which can squeeze through the alley. The cop cars... can't. Cue half a dozen cars all smashing into each other in a magnificent pile-up.
 * Double subversion in the movie Short Time: the main character engages in this sort of behavior during a car chase... because he is actually suicidal. However he ends up miraculously surviving and even gets a commendation!
 * The SFPD cops who make the unwise decision to try to chase James Bond (who is driving a stolen fire engine) in A View to a Kill.
 * The cops and FBI agents in Eagle Eye stupidly follow the protagonist through cross traffic at red lights, smashing into innocent civilians along the way. One even tries to ram him head-on!
 * Older than all of these are the Keystone Kops.
 * Partially Averted in The One, where the Police SUV's backtrack to avoid oncomming traffic on a bridge, and call in a Helicopter to help track the suspect. However, they still end up crashing at the end of the pursuit anyway.
 * This troper remembers watching part of a movie during a chase of a guy in a big rig. The cops are run off the road one by one, even after enlisting another big rig driver (who also crashes), and finally resorting to ramming the chased head-on with a school bus. The chased guy jumps out at the last minute, surviving.
 * Used in The Matrix series to almost comedic effect, while having a philosophical meaning at the same time, the Lemming Cops will always follow The Agents pursuing the heroes no matter what, being as reckless as The Agents are, causing much potential loss of human life as The Agents do, in their simple binary pursuit of the heroes, and even becoming, as needed, the vessels that The Agents use to continue to chase the heroes down.
 * Played for laughs in Beverly Hills Cop, as the police cars storming into the Big Bad's mansion to rescue the heroes get into a chain-reaction rear-end collision when a pair of Mooks crashes during an attempted getaway.
 * A rather pathetic example in Unstoppable where a bunch of cop cars make a tight turn to keep up with an out of control train, causing one of them to completely flip over in the process...despite the fact that there was absolutely no reason to pursue so recklessly (or at ALL), especially given that there was nothing they could do except hurt bystanders.

Live Action TV

 * The Dukes of Hazzard, almost every episode.
 * Lampshaded in an episode of Reno911 in an interview with Deputy Travis Junior, he mentions an incident when he was chasing drug runner "Fast Eddie" McLintock with about 15 other cruisers. The lead cruiser wrecked and started a pileup. Deputy Junior mentions that he saw the wreck about a mile up, as he was the last in line, but he just floored it, because how often do you get to be in a 15 car pileup?
 * Subverted in Blue Heelers when Jonesy pursues a suspect. When he is ordered to terminate the chase, he does (reluctantly), but does not acknowledge it to his superior, and when the suspect wraps his car around a tree Jonesy is very nearly brought up on disciplinary charges - the only thing that saves him is the discovery that the driver was probably on his mobile phone. Before he was chasing the suspect on a dirt road, when he couldn't see in the dust only to discover the suspect had stopped the (stolen, natch) car in the middle of the road, causing the police car to smash into it before flipping the bird and driving off.
 * And subverted again in several episodes before that. The most notable example would probably be Brad, Susie's ex husband, who became a paraplegic after a pursuit ended in a crash. He blames himself for what happened but listening to Brad and the pursuit controller he was at least as in control and unlemming like as most examples on this list, his only possible fault in hesitating to follow the order to terminate when he reports the suspect going 140 KPH (about 60 MPH) on a bad road.

Video Games

 * Cops in the Grand Theft Auto series are completely reckless in their pursuit of the player—they'll follow you off cliffs, into water, and plow through traffic, often causing explosive pile-ups. Sometimes the player's "Wanted" level will increase because so many cops have killed themselves in the chase. It gets even more ridiculous when, in Vice City, you turn on the cheat where going fast enough causes cars to fly
 * Literal Lemming cops in GTA4 here.. http://www.youtube.com/watch?v=xqzeZFO897A
 * Even on foot it goes sideways as many GTA police simply cannot swim.
 * In Grand Theft Auto: Chinatown Wars, besides escaping into a safehouse, the main/only way to get your wanted level to zero is to force the cops tailing you to crash, one at a time. While incredibly fun, this isn't the smartest mechanic, as it only counts when an occupied cop car crashes. Hitting an unoccupied one or killing a pedestrian officer will actually increase your wanted level.
 * The same also seems to, very ironically, apply to paramedics too - the ambulances seem to exhibit much the same behaviour as the cop cars, causing more fatalities than they prevent.
 * Need for Speed Most Wanted: Crashing into one of the marked buildings or roadside objects during pursuit will cause it to crush down on the police cars chasing you.
 * The cops are a lot stupider than just running into obstacles which they almost always can't avoid due to going too fast. They'll gladly run into their own roadblocks, sometimes opening a hole for the player they otherwise couldn't have achieved, and they will happily go zooming over the same jumps the player can, destroying their car in the process. Indeed, it sometimes makes the radio chatter feel full of Narm since they sound so much more intelligent on the radio than they actually act.
 * In Need for Speed Carbon, a cop car will often take up position in front of your car and stay there with remarkable precision. You can literally steer the cop car into oncoming traffic.
 * When the cops are after you in Driver, they seek to chase and ram you no matter at what costs, up to kamikaze-like head-on collisions when both you and the cops are driving at maximum speed. This is justified by the fact that nobody can get out of their car, and the squad cars are actually the police's only weapon against you. In the sequels, the cops are smarter, but still not smart enough not to chase you over ramps.
 * As with all the above examples, the cops in the Midtown Madness games are ridiculously single minded in their attempts to run the player off the road, including crashing through other traffic.
 * Scarface the World Is Yours. One essential missions involves a high speed chase with the cops literally flying off a high ledge to crash into the debris below and die horribly. Their 'crime'? Stealing donuts. So ... yeah.
 * Just Cause 2 gives us the notoriously thick Panauan military/police force. When on Rico's tail, they'll do anything to take him down, including crashing into each other and exploding, plunging off cliffs, blithely ignoring their burning vehicle that's due to explode in seconds, and sending streams of weak jeeps and motorcycles to challenge an armored vehicle.

Western Animation

 * Parodied in Family Guy when Stewie, Brian and Mort are in a submarine, being chased by another submarine. They manage to make their pursuer crash and the moment it does so, about two dozen police cars come in apparently from nowhere, crashing into the sub.
 * In an episode of The Simpsons Marge and her friend Laura Powers are being chased by the police across the desert. Homer, riding in Chief Wiggum's car (long story) successfully warns them that they're about to drive into The Grand Chasm and they brake to a stop in time, while Wiggum and Homer go hurtling past them and over the edge.
 * Parodied again in South Park, where Cartman and Kyle are riding their Big Wheels to California. A CHP Cruiser spots them and gives chase, at what could be no faster than 5 mph. Despite this, the Police Car somehow manages to lose control and flip over the median.
 * The Bikini Bottom police are prone to this type of pursuit, notably when chasing Spongebob and Mrs. Puff during the former's driving test. There is a tanker truck of fruit punch for seniors involved.

Real Life
The lady just smiled and waved. Oh, and this happened right in front of the police station.
 * A very mild case happened some years ago. A lovely lady was walking down the sidewalk. Three officers were coming up to a stop sign. The first officer stopped at the sign, and had a look at the eye candy. The other two officers had a look as well, but were Distracted by the Sexy and didn't stop...