All the stories pointed to research published in Diabetes Care in 2007 on body composition that used three different diets with the same calories but different nutrients. The researchers recruited 11 Spanish volunteers (that is right, only 11 subjects), average age of 62 years, with a family history of diabetes. All were obese and insulin-resistant. All subjects followed the diets for 28 days. One diet was high in carbohydrates; the second was high in saturated fat, and the third was a Mediterranean-type diet that was high in monounsaturated fats.
These heart-healthy fats are found in avocados, olives and olive oil, nuts and seeds and dark or semisweet chocolate. Results showed that weight and fat were unchanged, but when on the high-carbohydrate diet, body-fat distribution was altered and fat tended to accumulate in the gut. That is a bit different than claiming that foods high in monounsaturates shrink belly fat.