Fright Deathtrap

Murder Is the Best Solution; sadly, it's not always practical or possible to simply shoot the target. Either because they want to avoid being incriminated in the murder or have no corporeal body, the murderer will use a Fright Deathtrap to figuratively and literally scare someone to death. The Fright Deathtrap consists of scaring someone at just the right moment so they end up dead in circumstances that Occam's Razor will imply was an accident rather than foul play or the supernatural.

This usually requires a bit of set up, though; the murderer has to know the victim's routine, surroundings, and/or reactions well enough to manipulate them into a nearby danger, often one they have to identify or place themselves. For the sake of The Perfect Crime, it should be something that was always there, but a harmless gift (or abandoned object) serves just as well if it's in a place that can help kill the victim.

A few common variants:

Deadly Fall: Scaring a person at the top of a staircase so they fall and break their neck.
Run to your doom: Scaring a person into fleeing into a deadly trap, oncoming traffic, or another enemy. Often, they'd have lived if they simply stayed put or moved slowly.
Deer caught in headlights: Scaring a person motionless so that they ignore or fail to dodge or escape an approaching danger or falling object.
Scared Stiff: The victim is in such poor health, either mentally or physically, that the shock of a good scare itself may be enough to push a near-fatal condition over the edge.

This isn't anywhere near foolproof, of course, and runs the risk that failing to scare the victim into a Fright Deathtrap puts the thwarted murderer out in the open. If he could die a second time, a ghost might die of shame at being as ineffective as a Peek a Bogey Man. The would-be victim now knows someone (or thing) is out to get them. ...Unless the would-be murderer used a spring loaded cat as the scare. Then again, maybe the murderer was going for a long-term frightfest...

Examples of Fright Deathtrap include:

Anime and Manga

In Higurashi: When They Cry, during the last part of an arc, Mion is hiding under Keiichi's hospital bed, waiting for a chance to attack him when he's alone. But, much as it is a surprise to say, first, the REAL murderer was Shion, and second, in the first of the answer arcs, it's revealed that it was actually a hallucination that caused him to see the same thing over and over again until he finally died of a heart attack.

Comic Books

A standard tactic for the Scarecrow in the Batman comics.
Used in Ramba when she is hired to kill a mob boss and make it look like natural causes. She breaks into his doctor's office and learns that he has a weak heart. She then breaks into his bedroom and throws a knife at him. The knife is tied to a string around her wrist and stops short of his chest, but the fright triggers a fatal heart attack.
In the Golden Age, heroes occasionally did this, although usually unintentionally. The original Green Lantern (Alan Scott) terrified a villain into confessing that he had masterminded blowing up the bridge that Scott had designed, killing everyone on board the test train except Scott - and then the villain drops dead of a heart attack.

Film

In Ghost, the protagonist scares one of his murderers into running onto a busy street. The man gets hit by a car and then dragged to hell by living shadows. The hero still gets to ascend to heaven after disposing of the other murderer (who also died without direct action on Sam's part).
The Run to your doom variety is used frequently in Young Sherlock Holmes, as several elder gentlemen who pissed off the wrong Egyptian cult as younger men are drugged with blow-darts, causing them to see terrifying hallucinations and run into traffic, leap out 3rd story windows, etc.
In the film and play Death Trap, wife Myra Bruhl is literally frightened to death after witnessing a staged murder victim return from their grave. She collapses from a fatal heart attack, and the conspirators shake hands over the body.
The last variety occurs in Les Diaboliques.
In The Tingler, the theater owner's mute wife is frightened to death while alone in their apartment - hints suggest it may have been the work of coroner Vincent Price who may have 'medicated' her with LSD to get a 'scared to death' subject for his work but it turned out to be the work of her husband, caught red-handed with the spook show props that killed her.

Literature

Discworld: In a definite Crowning Moment of Awesome, a young Lord Vetinari does this to Lord Winder in Night Watch. The Properly Paranoid Winder is expecting to be poisoned or otherwise assassinated, and his nerves are so on edge that a simple "boo" causes him to die of fright.
How Sir Charles Baskerville was killed in The Hound of the Baskervilles.
Agatha Christie does this a few times, though she's just as likely to subvert it:
- In "The Blue Geranium", a woman is told by a fortune teller, "Beware the full moon. The blue primrose means warning, the blue hollyhock means danger, the blue geranium means death." At the next full moon, one of the primroses on her wallpaper turns blue, and at the full moon after that, one of the hollyhocks turns blue. The woman dies of a heart attack on the night of the third full moon, with the implication being that she was frightened to death by the threat. She was actually poisoned by her nurse, who switched her bottle of smelling salts with cyanide crystals. The nurse set up the whole blue flowers motif as camouflage.
- In "The Case of the Caretaker," a woman dies when her horse is frightened by the aforementioned caretaker, causing the animal to rear and the woman to fall off. Here again, the horse hadn't been frightened but rather had been shot by a bee-bee gun. And the woman died from poison rather than from the fall.
John Dickson Carr's locked room mysteries, which might be called "howdunnits", included a couple like this, where the mystery was largely just how the victims had been scared to their deaths.
- In The Case of the Constant Suicides, everyone who stayed in a certain room in a castle for a night would wind up falling down to their deaths from the dangerous balcony, as if something scared them into attempting to escape. There was nothing special in the room aside from a box with a cage door such as might be used to carry a small animal that had been brought in recently and left under the bed—but which people had looked into and found it to be empty. Actually it wasn't empty, but contained something nearly invisible -- carbon dioxide ice, which would start to vaporize as the temperature got lower at night, leaving the occupant of the room unable to breath and cause them to panic for some air.
- In He Who Whispers, just after it has been suggested that one of the characters is a vampire and was able to commit a previous impossible murder by flying, a shot is heard, and one character is found in her bed scared so badly she has nearly died (and is incapable of explaining what has happened, of course). She's holding a gun and appears to have shot at something ouside the window, which is, of course, so far above the ground and inaccessible that only something flying could have been behind it. The would-be-murderer -- who would have succeeded if he had had the right, much more sensitive target instead of the wrong person in the dark -- had in fact been in the room with the victim, pressed a gun to her head in the dark, and whispered to her a long time about how he was going to shoot her -- then fired the other gun he had towards the window, expecting her to die of shock when she thought she was being shot, but with it looking like she fired the gun herself.

Live-Action TV

Night Gallery episode "The Ghost Of Sorworth Place". A ghost appears near the top of a flight of stairs. A man pursuing the ghost tries to grab it but falls through it and down the flight of stairs, breaking his neck.
Attempted in one episode of Randall and Hopkirk Deceased by a ghost villain, who appeared in the middle of the road while his target was driving. But his target knew he was a ghost and just drove straight through him.
The Scared Stiff variant is attempted in an episode of Monk, where someone wants to keep the Worlds Oldest Man from reaching his next birthday.

Radio

The Shadow adores this trope. He uses his powers to cause hallucinations that make the villains kill themselves or their partners, or just freaks them out so badly that they're driven to do something suicidally stupid.
- In "The Three Ghosts" the villain is trying to do this to his wife, and apparently did it to his last one.

Tabletop Games

Torpedoes in Battlefleet Gothic often can be evaded by moving out of their way, but this turns them into a weapon for manipulation of enemy movement. Especially since harder manoeuvres require special orders, which reduce weapon performance and require a Command check, thus better not left until the last turn, and then the same applies to the repositioning move after it, if any. As such, evading a salvo can place the ships in a position giving further disadvantages -- it may turn a less-armored or less-armed side to the enemy, be separated from escorts, or be caught together with other ships in a Nova Cannon blast radius.

Video Games

Silent Hill starts this way (both the game and the movie). Harry swerves to avoid running over a woman in a schoolgirl uniform and crashes into Silent Hill. Tellingly, Harry fails to avoid her, and though she braces for the impact she shimmers like smoke after the jeep goes through her.
In Illbleed, many of the traps are meant to scare the target instead of physically injuring them. Your character will die of shock if their heart rate gets too high.
In an example of Run to your Doom, World of Warcraft has the infamous "Fear" spell available to Warlocks (And a myriad of other Fear inducing spells and abilities available to other classes)-- it finds its biggest use in PvP where it can be used to... well, in a way, scare the other player's character to death.
- Subverted if the feared object actually is a PvE monster running around and alerting nearby monsters so they join the battle, eventually killing the warlock (or whatever) and his whole group instead.
The Elysian Box in Professor Layton and the Diabolical Box seems to function this way. Rumors surrounding the box say that it kills all who dare open it, and the game begins with the death of Layton's old mentor while investigating the rumors it turns out that the box's "power" actually comes from the victim being exposed to a type of fume that causes the inhaler to be highly susceptible to suggestion to the point of perceiving whatever he thinks might be in the room as actually being there. Schrader actually recovers from his death at the end, but it's speculated that he may have been one of the lucky ones who wasn't say, chased out of a tenth story window by the imaginary thirty foot python lurking in the box, much like the Young Sherlock Holmes example.
In Dwarf Fortress creatures can dodge attacks (including ranged and/or trap weapon attacks), which puts them on an adjacent tile (including some normally inaccessible ones, such as statues). This became an important part of defensive design: in a simple variant, the enemies are either smashed into the pit by a huge spiked ball if it hits, or jump there trying to dodge it. This allows to construct traps effective against creatures who dodge too well or are resistant to your preferred trap weapons, and in case of enemies with ranged attacks, turn shoot-outs more in favour of the defenders. Let the enemies jump into another trap or a pit with Shark Pool or equivalent - though lava pool seems to be the most popular solution for those not interested in less heat resistant items or odd trinkets of goblin bones. On the other hand, you need to check where your crossbow squad is going to dodge when goblins shoot back. Another advantage is that traps that don't need to hit anything can be optimized to rarely hit and very rarely jam when they do (mediocre quality of mechanisms, training spears or other blunt weapons) — this keeps them working for much longer, and dwarves won't try to clean them while enemies are around.
- Some creatures don't dodge, however. Such as zombies.

Western Animation

The 1999's Disney short "How to Haunt a House," from House of Mouse: at the beginning, we hear Goofy getting hit by a car so that he can be a ghost and demonstrate how to haunt a house, with Donald Duck as the hauntee. After many amusing attempts that end in failure, he finally succeeds in scaring Donald, who runs out the door, is also hit by a car, and comes back inside as a rather angry ghost.