I have the sense that the doctrine of self-examination figures large in the writings of the Puritans (and perhaps other old-school Reformed theologians); however, I do not know nearly as much about it as I would like, as I am only at the very beginning of my study of the old Reformed theologians. What did the old Reformed theologians, particularly the Puritans, teach about the Christian practice of examining oneself—how it is to be done, its importance, what its purpose and result is, and where it arises from the Scriptures? What works by the Reformers and the Puritans might I turn to to study this teaching further? I would also be pleased if any answers wanted to include some addition information about what other Christian traditions have said about it.
I have read a lot of the Puritans and those that follow them and have as a result I do practice a lot more self examination. However, I have not found in my reading that this was necessarily a major topic for them. On the other hand I tend to find it threaded throughout their works in a more natural way. It was simply a part of the Christian life for them. There are however some specific works on the topic which Kazark has noted in his answer.
Among the Puritans I have personally found that the writings of Jeremiah Burroughs, Jonathan Edwards and Thomas Watson have had the greatest effect in driving me to self examination. Particularly Burrough's Gospel Fear and Gospel Conversation.
Their view of examination tends to be an encouragement with two prongs. First examine your heart. Do you love Christ. More specifically do you love the Christ of the Bible. Do you love the God who is both a consuming fire who will eternally damn sinners in hell as well as the loving Father who will save his Children from their well deserved fate.
Second, examine your actions. Does this love of God break forth in acts of love to the brethren, to the needy, and to all our neighbors.
So far I have found the following works, but have not had time to study them:
I also found these sermons, but have not studied them yet either:
I also intend to look up the relevant passages in Calvin's Institutes, his commentaries, and John Gill's and Matthew Henry's commentaries. I intend to report further...