Monumental Damage Resistance

The Washington Monument in the background is the dead give-away.

A major city has been completely leveled, whether by a marauding monster, invading alien, nuclear attack, or anything that has a tendency to cause massive collateral damage. The town is nothing but a smoking pile of smoldering rubble. But wait! What's that over there? A few buildings are still standing. And what do you know! It's The White House! It might be a little worse-for-wear, and probably has chunks taken out of it and bits broken off, but it's still standing and easily recognizable.

Basically, this trope is when widely recognized monuments are left remarkably whole in an otherwise devastated city. This occurs because, otherwise, no one would be able to tell just which city this wasteland used to be.

The inverse of Monumental Damage. Sometimes occurs after a Monumental Battle. A specific type of The Constant. It's often related to the Eiffel Tower Effect.

Example:

Anime and Manga

In Naruto, Pain's Shinra Tensei destroys the whole Konoha village, but does no damage to the Hokage mountain. It is later used to make Naruto realize whose village it is that was destroyed.

Film

In Resident Evil: Extinction, Las Vegas is buried in sand, but the monuments of the strip are still there and recognizable.
Independence Day creates a Funny Aneurysm Moment at one point in a shot of a devastated New York City. The World Trade Center is still standing, with only a few large chunks ripped out of it here and there.
Planet of the Apes: In the 40th century (the time period during which the first and second films take place), the Statue of Liberty lets the audience know that most of the movie takes place in a very post-apocalyptic New York.
Dreamscape. During the President's final dream, a train car is passing through a landscape blasted by a nuclear explosion. We see a partially destroyed Capitol building to let the audience know it's the ruins of Washington D.C.
In the movie version of Logan's Run, when Logan 5 and Jessica 6 reach the ruined city, we know its Washington DC because the Lincoln Memorial, Washington Monument, and National Archives are still standing. Vine-covered and weathered, but still standing.
The 1933 film Deluge, New York City is flooded. The Empire State Building is knocked down by the wall of water, but the Statue of Liberty remains standing. This makes this trope Older Than Television.
Played straight in The Day After Tomorrow, where pretty much every New York City landmark survives the flooding of the city and the subsequent hard freeze.
The 1956 film Earth vs. The Flying Saucers sees Washington, DC destroyed... again... and yet the White House is left standing.
Also played straight in 1975's Earthquake, which restricted most of the destruction to Los Angeles's outlying areas, leaving nearly all the city's landmarks fundamentally untouched.

Literature

The novel of Metro 2033 mentions that the Kremlin is standing and is in pretty good shape, compared to the rest of Moscow after the city was nuked. It is explained that NATO specifically avoided hitting Kremlin, preferring to test a new biological weapon instead. Said weapon consumed everyone holed up there and eventually became something much worse.

Live Action TV

One episode of The Outer Limits ends with a view of Washington DC destroyed. Inexplicably, the Washington Monument is still standing while everything around it is rubble.

Video Games

The intro-movie of Star Ocean: The Last Hope features the Earth consumed in the nuclear fires of World War III... and yet the White House and the Arc de Triomphe are both shown to be standing tall while their respective cities (Washington, DC, and Paris respectively) are flat fields of rubble.
Seen generally in the Fallout series, but most noticeable in Fallout 3. Despite the general devastation of the DC area, most of the landmarks are left remarkably intact. The Washington Monument, the National Archives, the Jefferson Memorial, the Museum of Natural History, and the Lincoln Memorial all feature prominently in in-game quests.
- Fallout 3 also subverts this trope. With so much of the city still standing, you'd expect to eventually be able to explore the White House, but when you get there, there's nothing but a huge blasted crater filled with blood-thirsty super-mutants where the White House once stood.
  - BLAST THOSE HUGE CRATERS! BLAST THEM ALL TO HELLLLLLLLL!
STALKER: Shadow of Chernobyl features the ferris wheel that still stands in the ghost town of Pripyat, which was based off of the exact same ferris wheel that was never used due to the disaster at Chernobyl. (and which has since collapsed in real life)
This trope is in full effect for the Washington DC levels in Modern Warfare 2. Both the Washington Monument and White House ("Whiskey Hotel") suffer damage but are still iconically standing. That said, they still take quite a beating, especially the White House, which gets quite a few walls knocked down and its interior utterly totaled.
- In Modern Warfare 3, though pretty much the entire New York City skyline is in flames, One World Trade Centre is conspicuously untouched. This one's pretty justified, though, for obvious sensitivity-related reasons.
Prominent at one point in the first Guilty Gear: Sol Badguy's stage takes place right next to the decapitated head of the Statue of Liberty.
No matter how many nukes you drop on city in Civilization IV, you can never destroy a World Wonder. Razing a city by conventional means will still destroy them, however.

Real Life

The Hiroshima Peace Memorial, aka the A-Bomb Dome, survived the atomic bombing despite being less than 500 feet away from the epicenter of the blast. It currently survives as a UNESCO World Heritage Site.
Cologne Cathedral - hit by seventy bombs, but the only building in the city to survive.
- Coventry Cathedral ... not so much.
There's an iconic photo from the London Blitz during 1940-1941 of St. Paul's Cathedral surrounded by smoke and fire from incendiary bombs falling all around it. However, while the area surrounding the Cathedral was practically wiped out, the Cathedral itself survived more or less unscathed.
- Similarly, one raid saw a bomb fall right on the Houses of Parliament, completely wiping out the House of Commons. The iconic clock tower housing Big Ben, however, was slightly damaged but otherwise okay.
A popular meme circulated after the earthquake and tsunami that struck Japan in 2011 shows two photos of a torii arch in Nagasaki (presumably the same one in both) standing alone and intact amidst first the aftermath of the atom bomb, and then amidst the wreckage of the tsunami, with the caption "What the hell is that arch made of?"