Osborn was educated at Ashburton High School, now known as Oasis Academy Shirley Park, Croydon, from 1983 to 1989. A central midfielder, Osborn started his career as a trainee at Crystal Palace in 1988 under the stewardship of Steve Coppell. He turned professional for the 1989–90 season, when his performances for a newly promoted Palace side secured their First Division survival with a 15th-place finish and saw him voted Young Player of the Year by the club's fans. However he missed out on a place in the squad for the final of the FA Cup that season, which Palace lost to Manchester United in a replay. They went on to finish third in 1990–91, one of the best finishes in the club's history. They then dropped to tenth in 1991–92, enough to secure them a place as founder members of the Premier League. They were relegated in 20th place in 1992–93, though had they have won their final game of the season then Oldham Athletic would have taken their place in the relegation zone. Osborn scored in the League Cup semi-finals in 1993, Palace's consolation in a 5–1 aggregate defeat to Arsenal. New boss Alan Smith led the "Eagles" to the First Division title in 1993–94, Osborn's last season at Selhurst Park.