Billy Wilder holds the record for writing more Best Picture nominees (7) than anyone else. Wilder's nominated and winning (marked with *) Best Picture films were:

• Ninotchka (1939)
• Hold Back the Dawn (1941)
• Double Indemnity (1944)
• The Lost Weekend (1945)*+
• Sunset Boulevard (1950)+
• Witness for the Prosecution (1957) - NOT screenplay-nominated
• The Apartment (1960)*+

+ - won Best Screenplay

Francis Ford Coppola and Alan Jay Lerner both hold the record for writing more Best Picture winners (3) than anyone else. Their nominated and winning (marked with *) Best Picture films were:

Francis Ford Coppola:

• Patton (1970)*+
• The Godfather (1972)*+
• The Conversation (1974)
• The Godfather: Part II (1974)*+
• Apocalypse Now (1979)
• The Godfather: Part III (1990)

Alan Jay Lerner:

• An American in Paris (1951)*+
• Gigi (1958)*+
• My Fair Lady (1964)*

+ - won Best Screenplay

African-American (Black) Screenwriters:

• Geoffrey Fletcher was the first African-American screenwriter to win an Oscar for screenwriting (adapted) for Precious (2009). It was his first produced screenplay.
• John Singleton, nominated for screenplay (original) for Boyz N the Hood (1991)
• Spike Lee, nominated for screenplay (original) for Do the Right Thing (1989)
• Charles Fuller, nominated for screenplay (adapted) for A Soldier's Story (1984)
• Lonnie Elder, nominated for screenplay (adapted) for Sounder (1972)
• Suzanne de Pass, co-nominated for screenplay (adapted) for Lady Sings the Blues (1972)
• Jordan Peele, nominated for screenplay (original) for Get Out (2017), his solo directorial debut film. [Note He was the first black filmmaker (and the third filmmaker of all time) ever nominated for directing, writing, and producing in the same year.]
• director Dee Rees, co-nominated for screenplay (adapted) for Mudbound (2017) [Note: the film's female cinematographer Rachel Morrison was also nominated for Best Cinematography - the first female to be nominated in the category!]

Best Picture Screenplay Writing Trivia:

Anita Loos was the second  woman to receive the sole screenplay credit for a Best Picture nominee with San Francisco (1936). Her play Gigi would later become the basis for the Best Picture-winning Gigi (1958). Joan Harrison became the first woman to co-author a Best Picture winner - Rebecca (1940) - with Robert E. Sherwood. [She also co-wrote Best Picture nominee Foreign Correspondent (1940) that same year (also a Hitchcock film.)] No woman has ever had the solo screenplay credit of a Best Picture winner. Ruth Prawer Jhabvala has the most solo screenplay credits (3) for a Best Picture nominee by a woman.

Wang Hui-Ling and Tsai Kuo Jung are the only Asian screenwriters (assisted by executive producer James Schamus) to write a Best Picture nominee - Crouching Tiger, Hidden Dragon (2000).

George Bernard Shaw was the only Nobel Prize winner to receive a Best Screenplay Oscar, for Pygmalion (1938) -- based on his own 1912 stage play of the same name.

Robert Riskin became the first writer to author two Best Picture winners with You Can't Take It with You (1938) - his previous Best Picture winner was It Happened One Night (1934). With My Fair Lady (1964), Alan Jay Lerner became the first writer to pen nominated screenplays for three Best Picture winners - his two wins were An American in Paris (1951) and Gigi (1958). His feat of three Best Picture winning screenplays has only been matched by Francis Ford Coppola.

Paul Haggis became the first screenwriter to have written two consecutive Best Pictures (with a Best Original Screenplay Oscar for Crash (2005) and a nomination for Best Adapted Screenplay for Million Dollar Baby (2004) (which lost to Sideways (2004))).

Dudley Nichols was the first Oscar winner to refuse an Academy Award - for his screenplay credit on The Informer (1935). He was boycotting the awards as a member of the Screen Writers Guild. (Two years later, after the Academy accepted the guilds and ended its support for union-busting activities, Nichols accepted his award.)

Woody Allen holds the record for most screenplay nominations (16, and all in the Best Original Screenplay category), but has only had three film screenplays which were also Best Picture nominees - Annie Hall (1977) (which won Best Picture), Hannah and Her Sisters (1986), and Midnight in Paris (2011). The 1977 and 1986 films gave Allen a Best Original Screenplay Oscar - two wins.

David Lean's two screenplays nominated for Best Picture were an incredible 38 years apart - Great Expectations (1946) and A Passage to India (1984).

Laurence Olivier's adaptation of Best Picture winner Hamlet (1948) was uncredited, making William Shakespeare the "official" writer of the film. Joseph L. Mankiewicz similarly gave credit for his screenplay of Julius Caesar (1953) to Shakespeare. Neither Olivier, Mankiewicz nor Shakespeare were given Best Screenplay nominations. Romeo and Juliet (1968) was the first Best Picture nominee directly adapted from Shakespeare that did not credit The Bard with the screenplay.

Frances Walsh, Philppa Boyens, and Peter Jackson's three screenplays that became Best Picture nominees were all from the Lord of the Rings trilogy. [Oddly, Stephen Sinclair was only involved with The Lord of the Rings: The Two Towers (2002).]

Albert Hackett and Frances Goodrich co-wrote the most Best Picture nominees (6) as a writing duo, including The Thin Man (1934)*, After the Thin Man (1936)*, It's a Wonderful Life (1946), Father of the Bride (1950)*, Seven Brides for Seven Brothers (1954)*, and The Diary of Anne Frank (1959). [Screenplay nominees are marked with *.]

All of Mario Puzo's screenplays for Best Picture nominees were from the Godfather trilogy. Goodfellas (1990) was the SOLE screenplay by Martin Scorsese to be nominated for Best Picture.

Best Picture champs usually win one of the two screenplay awards - approximately two-thirds have done so since 1950.

Best Picture Winners: Not Nominated for Best Screenplay Award

• Wings (1927/28)
• The Broadway Melody (1928/29)
• Grand Hotel (1931/32)
• Calvacade (1932/33)
• Hamlet (1948)
• The Sound Of Music (1965)
• Titanic (1997)
  

Best Picture Winners: Did Not Win Either a Best Screenplay Award or Best Director Award

• Wings (1927/28)
• The Broadway Melody (1928/29)
• Grand Hotel (1931/32)
• Mutiny on the Bounty (1935)
• The Great Ziegfeld (1936)
• Rebecca (1940)
• Hamlet (1948)
• All the King's Men (1949)
• Gladiator (2000)
• Chicago (2002)

Best Picture Winners: Won a Best Screenplay Award, But Did Not Win Best Director

• Cimarron (1930/31)
• The Life of Emile Zola (1937)
• An American in Paris (1951)
• The Greatest Show on Earth (1952)
• Around the World in 80 Days (1956)
• In the Heat of the Night (1967)
• The Godfather (1972)
• Chariots of Fire (1981)
• Driving Miss Daisy (1989)
• Shakespeare in Love (1998)
• Crash (2005)
• Argo (2012)
• 12 Years a Slave (2013)
• Spotlight (2015)
• Moonlight (2016)

Best Picture Winners: Won Best Director, But Did Not Win a Best Screenplay Award

• All Quiet on the Western Front (1929/30)
• Cavalcade (1932/33)
• You Can't Take it With You (1938)
• How Green Was My Valley (1941)
• Gentleman's Agreement (1947)
• Ben-Hur (1959)
• West Side Story (1961)
• Lawrence of Arabia (1962)
• My Fair Lady (1964)
• The Sound of Music (1965)
• Oliver! (1968)
• Rocky (1976)
• The Deer Hunter (1978)
• Platoon (1986)
• Unforgiven (1992)
• Braveheart (1995)
• The English Patient (1996)
• Titanic (1997)
• Million Dollar Baby (2004)
• The Artist (2011)
  

Ring Lardner Jr.'s screenplay of M*A*S*H (1970) bears little resemblance to the final film, but was the sole screenplay credit, winning the Best Screenplay Adaptation Oscar, widely seen as an "apology" for being blacklisted. Philip Dunne was given credit for The Robe (1953) as a front for blacklisted Albert Maltz. Ian McLellan Hunter was given credit for Roman Holiday (1953) as a front of blacklisted Dalton Trumbo. Michael Wilson's work on Friendly Persuasion (1956), The Bridge on the River Kwai (1957) (along with Carl Foreman) and Lawrence of Arabia (1962) went uncredited due to blacklisting, but each film earned an Oscar nominations for Best Screenplay Adaptation. All blacklisted writers were later officially re-credited for their work, and given their respective awards (often posthumously).