For my personal projects, where I'm the only developer (and in many cases the only user), I have a self-made issue tracker that I made years ago, when the only big name in (free)issue trackers was Bugzilla, which I didn't like.
For a new multideveloper project I'm working on, we're using Redmine. It's written in Ruby (on Rails) but it's easy to set up on Linux, supports a host of source control systems, has plugins and a wiki (including plugins to change the wiki syntax; I wrote one to do Mediawiki-style syntax since it was more familiar to the people using it). I don't know about it's API/integration but with plugins, odds are you can either find or write something to do what you need.