The original team's first mission involved them rescuing Orphan from her abusive captors. During the 1977 Gang War the group helped to restore order to city after Black Velvet's madness fell upon the city.
By 1995 a new group of young heroes took upon the name the Irregulars. In December of that year they battled with a massive creature of the Scavenger Peoples in a construction zone. The group kept up a presence through the late 1990s, helping Honor Guard defend Shadow Hill when an angry mob tried to burn it down in 1997 and helping defeat the Enelsians during the Enelsian Invasion. They fought with Longhorn who was part of the Conquistador's Plot in late 1998. In addition they teamed with Jack-in-the-Box at some point to travel into the future and stop The Wasting.
One of their last known entanglements was with Redline in April 2003, which caused a need for the group to be "recovered" from the Gaines River. They were likely still together in 2004, but there was a new Stray running around by the next year.
Goldenglove's mentioned reforming the Irregulars in 2016, strongly suggesting the group has been disbanded for a while.
Irregular is a military term for a soldier not a part of the regular army or who follows their rules.
Judging by Palmetto's reaction to Jailbait's reference to him as "Roach," in their first appearance, and the poster in Asa Martin's office, its possible that the modern group were just recently formed sometime in 1995.
It's unknown whether or not Goldenglove ever formally became a member of the group. In early 1999, she was planning on training with the team and recent dialogue would suggest she did have a connection to the group. In the Astro City: A Visitor's Guide, she wasn't pictured with the team.