• Monarch – Frederik IX^[1]
• Prime minister – Jens Otto Krag

• 10 September – Ole Olsen wins the 1971 Individual Speedway World Championship.

January–March edit

• 12 January - Peter Madsen, convicted murderer, former engineer and entrepreneur
• 7 February - Nicolai Wammen, politician
• 18 February - Thomas Bjørn, professional golfer
• 23 February - Nukâka Coster-Waldaum actress
• 25 February - Morten Wieghorst, football coach and former football player
• 13 March - Allan Nielsen, footballer

April–June edit

• 5 May
  • Helene Kirkegaard, badminton player
  • Anette Hoffmann, handball player
• 21 May – Kristian Jensen, politician
• 5 June
  • Rane Willerslev,anthropologist
  • Eske Willerslev, biologist
• 24 June – Thomas Helveg. footballer

July–September edit

• 23 August – Mads Fuglede, journalist and politician

October–December edit

• 30 November – Pia Olsen Dyhr, politician
• 31 December – Søren Pape Poulsen, politician (died 2024)

• 24 March – Arne Jacobsen, architect, furniture designer (born 1902)
• 1 May – Ejnar Mikkelsen, Polar explorer and author (born 1880)
• 6 October – Hans Christian Hagedorn, pharmacologist and Novo Nordisk co-founder (born 1888)
• 10 October – Hans Rasmus Hansen, politician (born 1896)

• 1971 in Danish television