These are the songs that reached number one on the Top 100 Best Sellers chart in 1961 as published by Cash Box magazine.
Issue date Song ArtistJanuary 7 Wonderland by Night Bert Kaempfert & OrchestraJanuary 14 Exodus Ferrante and TeicherJanuary 21 January 28 Will You Love Me Tomorrow The ShirellesFebruary 4 February 11 Calcutta Lawrence Welk & OrchestraFebruary 18 February 25 March 4 March 11 Pony Time Chubby CheckerMarch 18 Surrender Elvis PresleyMarch 25 April 1 Blue Moon The MarcelsApril 8 April 15 April 22 Runaway Del ShannonApril 29 May 6 Mother-in-Law Ernie K-DoeMay 13 May 20 Runaway Del Shannon May 27 Mother-in-Law Ernie K-DoeJune 3 Running Scared Roy OrbisonJune 10 Travelin' Man Ricky NelsonJune 17 June 24 July 1 Quarter To Three Gary U.S. BondsJuly 8 July 15 July 22 Tossin' and Turnin' Bobby LewisJuly 29 August 5 August 12 August 19 Michael The HighwaymenAugust 26 September 2 September 9 September 16 Take Good Care Of My Baby Bobby VeeSeptember 23 September 30 October 7 Crying Roy OrbisonOctober 14 Hit the Road Jack Ray CharlesOctober 21 Runaround Sue DionOctober 28 November 4 Big Bad John Jimmy DeanNovember 11 November 18 November 25 December 2 December 9 The Lion Sleeps Tonight The TokensDecember 16 December 23 December 30
See also 1961 in music List of Hot 100 number-one singles of 1961 (U.S.)
References https://web.archive.org/web/20110610234634/http://cashboxmagazine.com/archives/60s_files/1961.html