Does Every Modern Doctor Who Feature a Hidden or Crashed Spaceship?

No, but a surprising number from the modern series. I count 31 firm "yesses," and a lot of maybes.

Eccleston

"Rose": Alien species on Earth with a hidden transmitter.
"The End of the World": No, just creepy skin person.
"The Unquiet Dead": No, but alien race hiding on Earth.
"Aliens of London": Yes. (Also hidden aliens.)
"World War Three": Yes. (Also hidden aliens.)
"Dalek": No, unless the Dalek is a spaceship. (Dalek is hidden.)
"The Long Game": No, but hidden alien.
"Father's Day": No. Even the TARDIS goes missing.
"The Empty Child": Yes. (Hidden nano-aliens.)
"The Doctor Dances": Yes. (Hidden nano-aliens.)
"Boom Town": Yes. (Hidden alien.)
"Bad Wolf": Yes. (Dalek ships around Earth. Hidden aliens!)
"The Parting of the Ways": Not exactly at this point, but continuation of previous episode.

Tennant

"The Christmas Invasion": No. Straight out invasion.
"New Earth": No. (Hidden human!)
"Tooth and Claw": Not precisely. An alien species seems to have come to Earth. In a hidden fashion.
"School Reunion": No, just hidden aliens attempting to conquer time and space.
"The Girl in the Fireplace": Yes. (Hidden robots.)
"Rise of the Cybermen": No.
"The Age of Steel": No.
"The Idiot's Lantern": No, just a hidden alien.
"The Impossible Planet": No. Hidden devil.
"The Satan Pit": No. Hidden devil.
"Love & Monsters": No, just a hidden alien.
"Fear Her": Yes. (Hidden alien.)
"Army of Ghosts": Kind of. The hidden spaceship (and aliens) is in the Void.
"Doomsday": Not precisely.

"The Runaway Bride": Yes! Both in orbit and (kind of) at the heart of the plant, though it's more appropriately an incubation pod. (Hidden alien!)
"Smith and Jones": No, although hospital is transported to the moon.
"The Shakespeare Code": No. (Hidden aliens.)
"Gridlock": No, just enormous hidden crabs.
"Daleks in Manhattan": Not exactly: hidden Daleks.
"Evolution of the Daleks": Same.
"The Lazarus Experiment": No.
"42": No. (Alien hidden in brains?)
"Human Nature": Yes. (Aliens hidden in jars.)
"The Family of Blood": Yes. (Yes.)
"Blink": No, just hidden aliens.
"Utopia": No. (Unless you count each crazy end-of-universe person as a spaceship. They are hidden humans, not really aliens.)
"The Sound of Drums": No.
"Last of the Time Lords": No.
"Voyage of the Damned": Not exactly, though the ship does show up around Earth, hidden.

"Partners in Crime": Kind of. (The ship comes back, but the whole building is full of alien tech. And there are hidden aliens.)
"The Fires of Pompeii": Yes. (Hidden stone aliens.)
"Planet of the Ood": No. (Hidden brain.)
"The Sontaran Stratagem": Yes, and hidden Sontarans.
"The Poison Sky": Ditto.
"The Doctor's Daughter": No.
"The Unicorn and the Wasp": No, but a hidden alien.
"Silence in the Library": No — and they're not aliens, either; they're native.
"Forest of the Dead": No.
"Midnight": No. (Hidden non-alien.)
"Turn Left": No. (Phase-shifted hidden alien.)
"The Stolen Earth": Sort of? The Dalek ship is hidden initially.
"Journey's End": No.

"The Next Doctor": Yes — sort of? (Hidden Cybermen.)
"Planet of the Dead": No.
"The Waters of Mars": No. (Hidden virus.)
"The End of Time": Yes.

Smith

"The Eleventh Hour": No, just a hidden alien.
"The Beast Below": No, just a hidden space whale.
"Victory of the Daleks": Yes.
"The Time of Angels": Yes.
"Flesh and Stone": Yes.
"The Vampires of Venice": No? I guess just hidden aliens.
"Amy's Choice": No.
"The Hungry Earth": No — they were already there and not aliens!
"Cold Blood": No.
"Vincent and the Doctor": No, just a hidden alien.
"The Lodger": Yes. But no aliens, just holograms.
"The Pandorica Opens": No.
"The Big Bang": No.
"A Christmas Carol": No.

"The Impossible Astronaut": No? There's a whole control system underground for the hidden aliens, but I guess not a spaceship as such.
"Day of the Moon": No?
"The Curse of the Black Spot": Yes.
"The Doctor's Wife": No, although I guess the whole intelligent planetoid is a…ship? That they were unaware of?
"The Rebel Flesh": No.
"The Almost People": No.
"A Good Man Goes to War": No.
"Let's Kill Hitler": No.
"Night Terrors": No, just a hidden alien.
"The Girl Who Waited": No.
"The God Complex": Yes. (Alien isn't precisely hidden.)
"Closing Time": Yes!
"The Wedding of River Song": Yes! The Teselecta!!
"The Doctor, the Widow and the Wardrobe": Yes.

"Asylum of the Daleks": Yes! And then Clara is also a hidden "alien" among Daleks.
"Dinosaurs on a Spaceship": No — wait, yes! There's a hidden craft inside the bigger craft.
"A Town Called Mercy": Yes.
"The Power of Three": Yes.
"The Angels Take Manhattan": Not exactly — more like a kind of prison?
"The Snowmen": No, and is the Great Intelligence an alien?

"The Bells of Saint John": No.
"The Rings of Akhaten": No, just a hidden alien.
"Cold War": No, just a hidden alien.
"Hide": No, just a hidden alien.
"Journey to the Centre of the TARDIS": No.
"The Crimson Horror": Yes. (Rocket as opposed to a spaceship. Hidden worm thing that's native to Earth, I guess.)
"Nightmare in Silver": Mmmm, hidden Cybermen, but not precisely a ship.
"The Name of the Doctor": No, unless you count the TARDIS.
"The Night of the Doctor": No.
"The Day of the Doctor": No. (Hidden Zygons!)
"The Time of the Doctor": No.

Capaldi

"Deep Breath": Yes, with disguised robots. (And Missy may also be on a hidden ship.)
"Into the Dalek": Yes! The ship is hidden from the Daleks!
"Robot of Sherwood": Yes, with disguised robots.