Some stats:

Number of nominations per year:
  • 1940: 15
  • 1941: 13
  • 1942: 14
  • 1943: 11
  • 1944: 12
  • 1945: 13
  • 1946: 15
  • 1947: 10
  • 1948: 12
  • 1949: 13

Directors with multiple nominations:

2
Akira Kurosawa: Drunken Angel; Stray Dog
Ben Sharpsteen: Dumbo; Pinocchio
Clarence Brown: National Velvet; The Yearling
David Lean: Brief Encounter; Great Expectations
Edward F. Cline: My Little Chickadee; The Bank Dick
Elia Kazan: A Tree Grows in Brooklyn; Gentleman's Agreement
Frank Capra: Arsenic and Old Lace; It's a Wonderful Life
Mervyn LeRoy: Little Women; Waterloo Bridge
Robert Hamer: Dead of Night; Kind Hearts and Coronets
Robert Siodmak: The Killers; The Spiral Staircase

3
Billy Wilder: Double Indemnity; The Lost Weekend; The Major and the Minor
Ernst Lubitsch: Heaven Can Wait; The Shop Around the Corner; To Be or Not to Be
George Cukor: A Woman's Face; Adam's Rib; Gaslight; The Philadelphia Story
John Huston: Key Largo; The Maltese Falcon; The Treasure of the Sierra Madre
Michael Curtiz: Casablanca; Mildred Pierce; Yankee Doodle Dandy

4
Jacques Tourneur: Cat People; I Walked with a Zombie; Out of the Past; The Leopard Man
John Ford: How Green Was My Valley; My Darling Clementine; The Grapes of Wrath; They Were Expendable
Michael Powell and Emeric Pressburger: A Matter of Life and Death; Black Narcissus; The Life and Death of Colonel Blimp; The Red Shoes
Orson Welles: Citizen Kane; The Lady from Shanghai; The Magnificent Ambersons; The Stranger
William Wyler: Mrs. Miniver; The Best Years of Our Lives; The Heiress; The Little Foxes

5
Michael Powell: A Matter of Life and Death; Black Narcissus; The Life and Death of Colonel Blimp; The Red Shoes; The Thief of Bagdad

6
Howard Hawks: His Girl Friday; I Was a Male War Bride; Red River; Sergeant York; The Big Sleep; To Have and Have Not
Preston Sturges: Hail the Conquering Hero; Sullivan's Travels; The Lady Eve; The Miracle of Morgan's Creek; The Palm Beach Story; Unfaithfully Yours

9
Alfred Hitchcock: Foreign Correspondent; Lifeboat; Notorious; Rebecca; Rope; Saboteur; Shadow of a Doubt; Spellbound; Suspicion