  I've started reading Psalm 56 this week in preparation for this week's Sunday School class. I'm not ready to post an outline yet, but early returns so far suggest that it's going to be a lot of fun.
The Psalm's prescript connects it with "When David was captured by the Philistines in Gath", which occurs in I Sam 21:10-15. The Psalm is David's expression of trust in God while experiencing fear of his enemies, and a call for God to put them down.
The fun part is going to be harmonizing the Psalm with the I Sam passage. In that passage, when David is brought before Achish, King of the Philistines, he pretends to be insane, scribbling on the walls of the city and drooling in his beard. Achish sends him away, because he says he has enough crazies around. David is thus spared by his deception from being killed or enslaved. Here's the problem then- if this story is designed to fit with our nice traditional value morals, it has to condemn David for his lies and deceptions and lack of faith in God.
But if it does that, then Psalm 56 stinks to heaven of hypocrisy and is useless for Christians. There must be something bigger going on in the story, and if Psalm 56 is inspired and useful for us, we must interpret that story in a way that leads to David's vindication. More to come. 
