But midfielder Gary Fraser has signed a new contract until January.
Manager Alan Archibald said of the 22-year-old, who has not played this season: "Gary has had a horrendous time with injuries over the last two years.
"And we want to give him the chance to prove his fitness and fight for his place in the team."
Fraser, who signed from Bolton Wanderers in 2014, has not played since April last year.
"He has plenty of football ahead of him and we are hopeful he can put the last 18 months behind him and hit the ground running when we return for pre-season," Archibald told his club website.
The manager pointed out that Thistle handed Welsh a similar short-term deal two years ago before the midfielder returned to the first team.
Archibald paid special tribute to Welsh despite deciding against offering a new contract to the 27-year-old, who signed from Hibernian in 2012.
"Sean has been at the club for more than five years and has been a vital part of the squad throughout that time," he said.
"He was instrumental in our promotion bid and was my captain for the first two seasons in the Premiership, playing a massive part in us securing top-six last month.
"We are in a much better position than in previous years as we have the majority of this year's squad under contract for next season and, with a number of those players in Sean's position, it means that there isn't the opportunity to offer Sean the contract he would want."
Welsh has made 28 appearances this season, scoring four goals, as Thistle finished sixth in the Scottish top flight.
Englishman Amoo joined Thistle in 2015 after leaving Carlisle United and the 26-year-old has played 31 times this season.
Ridgers has made only one start and one substitute appearance since the 26-year-old signed from Kilmarnock last summer.
Wilson made four appearances for Thistle at the start of the season, but the 22-year-old ended the campaign helping Stranraer avoid relegation from League One while on loan.
Meanwhile, central defender Niall Keown has returned to parent club Reading after playing 16 times while on loan from the English Championship club.
Former captain Sean Welsh, fellow midfielders David Amoo and David Wilson, plus goalkeeper Mark Ridgers, are being released by Partick Thistle.