Sheedy started his career with Hereford United, followed by a short spell with Liverpool, but played just three competitive games in four years and was sold to Everton for £100,000 in 1982. At Everton he made 357 appearances (12 as substitute) and scored 97 goals.
Sheedy's most notable achievements were as part of the 1984 Everton team that won the FA Cup (although he missed the final through injury). He was part of Everton's title winning teams in 1985 and 1987, and in 1985 the European Cup Winners' Cup, scoring in the final itself. In the 1980s Sheedy scored the most goals from free-kicks in the top-flight of the English football league. Perhaps his most famous free-kick moment game against Ipswich in 1985 when he scored with a 25-yard free-kick into the goalkeeper's left-hand corner, but having been forced to re-take the kick, proceeded to curl the ball into the keeper's right-hand corner. He was selected in the PFA Team of the Year in both Everton championship winning years of 1985 and 1987.
Though born in Wales, Sheedy was eligible to play for the Republic of Ireland as his father was from County Clare. He played 46 times for Ireland scoring nine goals, including one in a game in the 1990 World Cup against England. Ireland drew all three of their group games to qualify for the last 16 of the tournament, and subsequently beat Romania 5–4 on penalties, Sheedy scoring the first of the Republic's penalties. The Republic went on to lose 1–0 to hosts Italy in the Quarter Final of the FIFA World Cup. In this match, such was Sheedy's threat that Italy put the legendary Giuseppe Bergomi to man-mark Sheedy.
Sheedy was also part of the Euro 88 squad and played all 3 matches. He is also noted as the first-ever Republic of Ireland player to score a goal in the World Cup finals.