Mr. Starship

This trope describes a situation in a work of fiction, in which a prominent vehicle used by characters in the story is itself endowed with the qualities of being a character in the eyes of the audience. This is especially probable if said vehicle is essential to the advancement of the plot of the work or the actions of the main characters. Such endowment is represented by sorrow or grief in the event of the destruction of said vehicle. See Spaceship Girl for when the ship is literally a character.

Examples of Mr. Starship include:

Anime

The Mobile Suits of Mobile Suit Gundam Wing, especially in Mobile Suit Gundam Wing Endless Waltz. Manly Tears were shed during the finale when the Gundams self-detonated.

Film

Back to The Future Franchise - The destruction of the DMC time machine at the end of the third film has been known to cause some viewers to shed a tear or two.
Star Trek III: The Search For Spock - When the original Enterprise burned up over Genesis, Star Trek fans cried the world over.
Star Trek Generations - Essentially the same thing with the Enterprise D. That's what happens when you let Troi drive.
Chitty Chitty Bang Bang - This instance of the trope is not so much the fault of the audience as it is the fault of the main characters who endow Chitty as a character themselves. One may notice that the characters only take it so far; for example, they never refer to Chitty as 'he,' only as 'it.'
This has been true of the TARDIS in Doctor Who for years. You can actually see a gradual process of treating it as a simple "capsule" to the Doctor's longest, truest companion over the length of the series. Then, finally, we got to meet her, and it was one of the most tear-jerking, heartwarming episodes ever.

TV

Airwolf - The Airwolf Helicopter played such a crucial role in the series that even the main characters routinely endowed it and endeared it as a character all it's own.
Firefly - If Serenity didn't exist, then those space cowboys would have nowhere to go. The characters themselves treat her as part of the crew, as does director Joss Whedon.
Supernatural - The Winchester's Impala. So much so that the framing story of the season 5 finale was the biography of that car, treating it as the most important thing in all creation.

Tabletop Game

Rogue Trader supports it in mechanics: in addition to choice of hardware and crew training level, PC and NPC ships have quirks or complications - normally, one roll on Past Histories table and one on Machine Spirit Oddities (such as "Martial Hubris" or "Stoic").

Web Original

This is often literally true of spacecraft in Fenspace -- even if handwavium didn't create a governing Artificial Intelligence outright, the inevitable quirking results in ships with unique "personalities".