I agree with dinonum. One can learn from their mistakes without regret. IMO regret is nothing more than humble condemnation. The Bible tells us there is no condemnation in Christ. There is conviction, but not condemnation. Conviction motivates someone to repent. Condemnation pushes away. Jesus never pushed anyone away. He led people to follow Him, and He taught them through gentleness and conviction. Not condemnation. Regret, imo, is a mild form of condemnation.
I also think it stems from an excessive focus on self. "I wish I hadn't done that!"; "If only I hadn't made that choice!"; "If I could only go back and redo it!"; "What would my life be like now if I had made a different choice?" Instead of focusing on ourselves, are are to focus on God, and through Him, we are to serve others. Regret may eventually bring about "change", but imo, it's a selfish change. I say that because we should never regret the lessons God teaches us. Granted, the trials may have been difficult, and yes we may have to live with some painful consequences, but is it better to wish you could go back and change it, or to REALLY learn from the mistake and move on and do better? Regret focuses on what you can not change, instead of moving forward to what you can.
Now, I think there's a difference between a momentary wish of "Man that was a dumb decision, I wish I'd never done it", and a consistent state of living in regret. I guess my point is, that everything happens for a reason, and even if the things that happened were horrible, there's always something good to learn from it. So, imo, we shouldn't regret things because they shape us into who we're meant to be.