Prior to the acknowledgment of an insurgency in Iraq and the need for a new strategy, COIN rested in the capable hands of unconventional troops. For a look at a successful COIN campaign, visit Joint Special Operations Task Force - Philippines blog here. For a look at an unsuccessful one, google "Iraq".
You can read about forcing a square peg into a round hole in The New Republic article, "Course Correction". This serves as a prime example illustrating that theory and practice are two very different things.
My favorite quote from the article:
"We’re not trained to do governance and development," one young officer said. Another chimed in: "Shouldn’t the State Department and others be doing this? This isn’t our job."