Glad to oblige:
1. Cloverfield (The statue is decapitated by a monster)
2. The Day after Tomorrow (The statue is hit by a tidal wave and subsequently frozen in ice)
3. The Planet of the Apes (Remains of the statue on a beach)
4. Flodder in America (Anti-social Dutch family accidentally blows up the head of the statue)
5. Splash (Mermaid Madison comes ashore at the foot of Lady Liberty)
6. Remo: Unarmed and Dangerous (Remo walks on liquid cement during a fight at the foot of the statue).
7. X-men (A fight between the X-men and other mutants takes place on the statue).
8. The Godfather (Vito Corleone first sees the statue from behind bars in the holding centre of Ellis Island)
9. Ghostbusters II - The statue comes to life and wades into Manhattan.
10. Titanic - Rose (Kate Winslet) sees the statue as she's arriving back in the US.
11. Saboteur (Alfred Hitchcock) - A confrontation between Robert Cummings and villain Norman Lloyd in the torch of the Statue of Liberty
12. Funny Girl (I am pretty sure, someone can correct me if I am wrong.)
13. Les Triplettes de Belleville (Although a cartoon, the statue holding an icream cone and a hambuger bun is surely a representation of lady liberty).
14. Meteor (The meteor that strikes New York passes right by the statue of liberty, without damaging it).
15. National Lampoon's European Vacation
16. "The Brain" (Le Cerveau) The climax of this movie involves a huge duplicate of the Statue of Liberty.
17. The opening shot in Working Girl.
18. Team America, world police (The team flies past the Statue of Liberty on their way to the final showdown)
19. The Return of Captain Invincible (Superhero lives in statue)
20. A Journey for Margaret (Traumatised little girl, parents lost in London war-time bombings, is adopted by American couple and sails to America. As they arrive in New York, passing the Statue of Liberty, the lights of the city start to go out, symbolising America's entry into WWII.)
And I thought of another topic that I quite like,
desert islands. There should be quite many films set on quasi uninhabited islands, i.e. the main part of the plot is set on an island with (at least initially) no known inhabitants. I can't remember if this was already a topic once, but so what. Starting shot:
1. The Last Island (1990, a feminist version of the castaway theme)
Let's hear it...