These are the songs that reached number one on the Top 100 Best Sellers chart in 1960 as published by Cash Box magazine.
Issue date Song ArtistJanuary 2 Why Frankie AvalonJanuary 9 January 16 January 23 Running Bear Johnny PrestonJanuary 30 February 6 February 13 Teen Angel Mark DinningFebruary 20 February 27 Theme from "A Summer Place" Percy Faith & OrchestraMarch 5 March 12 March 19 March 26 April 2 April 9 April 16 April 23 Stuck On You Elvis PresleyApril 30 May 7 May 14 May 21 Cathy's Clown The Everly BrothersMay 28 June 4 June 11 June 18 June 25 Everybody's Somebody's Fool Connie FrancisJuly 2 July 9 July 16 Alley Oop The Hollywood Argyles, Dante & the EvergreensJuly 23 I'm Sorry Brenda LeeJuly 30 August 6 Itsy Bitsy Teenie Weenie Yellow Polka Dot Bikini Brian HylandAugust 13 It's Now or Never Elvis PresleyAugust 20 August 27 September 3 September 10 The Twist Chubby CheckerSeptember 17 September 24 October 1 October 8 My Heart Has a Mind of Its Own Connie FrancisOctober 15 Save the Last Dance for Me The DriftersOctober 22 October 29 November 5 November 12 November 19 November 26 Are You Lonesome Tonight? Elvis PresleyDecember 3 December 10 December 17 December 24 December 31
See also 1960 in music List of Hot 100 number-one singles of 1960 (U.S.)
References https://web.archive.org/web/20110818052118/http://cashboxmagazine.com/archives/60s_files/1960.html