• Monarch – Christian IX^[1]
• Prime minister – J. B. S. Estrup

Undated edit

• The Nordic Exhibition of 1888 took place in Copenhagen.^[2]

Music edit

• 5 February – First public performance of Carl Nielsen's String Quartet No. 1 in the smaller hall of the Odd Fellows Mansion in Copenhagen.

January–March edit

• 6 March – Hans Christian Hagedorn, pharmachologist and Novo Nordisk co-founder (died 1971)
• 9 March – Ernestine Nyrop, textile artist (died 1975)
• 15 March – Sophus Nielsen, footballer (died 1963)
• 30 March – Einar Utzon-Frank, sculptor (died 1955)

April–June edit

• 26 April – Olaf Henriksen, Major League baseball player (died 1962)

July–September edit

• 15 July – Svend Bille, actor (died 1973)
• 28 July – Adam Fischer, sculptor (died 1968)
• 12 August – Prince Axel of Denmark (died 1964)

October–December edit

• 13 December – A. J. Iversen, cabinetmaker and furniture designer (died 1979)
• 15 December – Kaare Klint, architect (died 1954)
• 16 December – Ivan Joseph Martin Osiier, Olympic athlete (died 1965)

• 29 February – Wilhelm Sponneck, nobleman and politician, Danish finance minister (born 1815)
• 8 August – Jørgen Roed, painter (born 1808)
• 14 August – Carl Christian Hall, politician, Danish prime minister (born 1812)