Rangers played a total of 22 competitive matches during the 1895–96 season.
They finished second in the Scottish League Division One with a record of 11 wins from 18 matches.
The club ended the season without the Scottish Cup after being knocked out in the quarter-final stage by Hibernian by 3–2.
The club had beaten Dumbarton and St Mirren during the competition.
All results are written with Rangers' score first.