I would list many of the same ones from above.
However, my FAVORITE film (classic or otherwise) is:
Some Like It Hot - 1959, Marilyn Monroe, Tony Curtis, Jack Lemmon (directed by Billy Wilder). When two musicians witness a mob hit, they flee the state in an all female band disguised as women, but further complications set in.
Dead Ringer - 1964, Bette Davis, Peter Lawford, Karl Malden (directed by Paul Henreid). The twin sister of a callous wealthy woman murders her out of revenge and assumes the identity of the dead woman. Bette Davis plays both roles. Fabulous.
Now, Voyager - 1942, Bette Davis, Claude Rains, Paul Henreid (directed by Irving Rapper). Boston spinster blossoms under therapy and finds impossible romance.
To Catch A Thief - 1955, Cary Grant, Grace Kelly, Jessie Royce Landis (directed by Alfred Hitchcock). When a reformed jewel thief is suspected of returning to his former occupation, he must ferret out the real thief in order to prove his innocence.
The Birds - 1963, Rod Taylor, Tippi Hedren, Suzanne Pleshette (directed by Alfred Hitchcock). A wealthy San Francisco socialite pursues a potential boyfriend to a small Northern California town that slowly takes a turn for the bizarre when birds of all kinds suddenly begin to attack people there in increasing numbers and with increasing viciousness.
Pollyanna - 1960, Jane Wyman, Hayley Mills (directed by David Swift). A young girl comes to an embittered town and confronts its attitude with her determination to see the best in life.
The Parent Trap - 1961 - Hayley Mills, Maureen O'Hara (directed by David Swift). Hayley Mills plays twins who, unknown to their divorced parents, meet at a summer camp.
True Stories - 1986, David Byrne, John Goodman (directed by David Byrne). A small but growing Texas town, filled with strange and musical characters, celebrates its sesquicentennial and converge on a local parade and talent show. (quirky and fun!)