McStay signed for Celtic aged sixteen from Celtic Boys Club in 1981. He made his first team debut for Celtic on 23 January 1982 in a 4–0 win over Queen of the South in the Scottish Cup. A week later on 30 January, he made his first league appearance in 3–1 win over Aberdeen at Pittodrie. He scored the third goal, taking a pass from George McCluskey, running through the Aberdeen defence and strking a left foot shot past Jim Leighton in goal.
Season 1982-83 saw the young midfielder establish himself as a first team regular, and he picked up his first winners medal on 4 November 1982 in Celtic’s 2–1 win over Rangers in the League Cup Final. His performances resulted in media speculation that Inter Milan were considering making a £2 million bid to take him to Italy, a rumour that was emphatically dismissed by the Celtic Chairman, Desmond White.
McStay continued to progress at Celtic, and he scored Celtic’s equaliser in the 1984 Scottish Cup Final against Aberdeen to take the match into extra time. However, Celtic played most of the match with 10 men after Roy Aitken had been sent off in the first half, and Aberdeen scored in extra time to win 2–1.
In December 1987, during Celtic’s centenary season, McStay signed a five-year contract at Celtic. He went on to enjoy his finest season, winning both the SPFA and Scottish Football Writers player of the year awards as the club clinched a League and Cup double in 1988.
When Roy Aitken left Celtic Park in 1990, McStay was appointed club captain, a position he retained until his retirement following the 1996–97 season. In his time with the club, Celtic won the League title three times, the Scottish Cup 4 times and the League Cup once. Although the second half of McStay’s career coincided with a time when Celtic were in turmoil and were overshadowed by rivals Rangers, in 2002 he was voted a member of Celtic’s greatest ever team by the club’s fans. He is also a member of the Scotland Football Hall of Fame, which honours the best players to play in Scotland and is located in the Scottish Football Museum.
McStay’s great-uncles, Jimmy and Willie McStay, were former Celtic captains, and his brothers Willie and Raymond also played for Celtic. His nephew, John, played with Celtic Boys Club before moving onto Motherwell under 19s and now plays for Ayr United as a defender.