Ablett was born in the Liverpool suburb of Aigburth. He played for Liverpool between 1985 and 1992. With the club, he won two league titles and an FA Cup, winning against Everton in the 1989 final. In 1995, Ablett moved from Liverpool to city rivals Everton. Later that year, he won the FA Cup with Everton. He remains the only played to win an FA Cup with both clubs. He also played for Derby County, Hull City, Sheffield United, Birmingham City, Wycombe Wanderers, Blackpool and the Long Island Rough Riders during his career. He was also a youth team manager for Liverpool and Everton. From 2009 to 2010, he was manager of Stockport County.