I usually find I’m using whatever the relevant company/client is tied into with other software integrations or the IT guy has done a deal with.
Current one is using Jira. It’s okay. Most I’ve used are “okay”. I think to get best out of it you want various plugins and integrations. They’ve got HipChat also but it’s not really used as there’s a preference for Skype, which doesn’t provide team chat in my opinion, at least not until Skype for Teams comes out next year.
Another option is Team Foundation Server, plus you get cloudy version for limited seats included in Visual Studio Online that you get with your VS account. The online stuff has Agile stuff and can integrate with Git instead of TFS.
GitHub has an issue tracker also, though fairly simple.
I’ve used StarTeam in the past. Horrible.
Bugzilla is free (with pay for premium support option) and quite popular, if a bit ugly and not easy to use. Though lots of customisation and integration options.
as an aside we used to use SourceSafe for code, but are moving to GIT
Good move. SourceSafe is awful! 😮 Even moving to svn is preferable to SS.
You’ll have to get your head round terminology differences though. Check-out in SS means to lock it for your use to modify. Git, svn and other’s check-out means download a working copy you can dick around with. Nothing is locked. You commit (and merge) changes when ready.