Osgood was founded in the 1880s. While its earliest settlers were primarily Protestants, a Roman Catholic parish was erected in the village in 1906. Today, St. Nicholas’ Catholic Church has been recognized as a historic site.
According to the United States Census Bureau, the village has a total area of 0.34 square miles (0.88 km2), all of it land.
As of the census of 2010, there were 302 people, 120 households, and 82 families residing in the village. The population density was 888.2 inhabitants per square mile (342.9 /km2). There were 121 housing units at an average density of 355.9 per square mile (137.4 /km2). The racial makeup of the village was 98.3% White and 1.7% Native American. Hispanic or Latino of any race were 1.3% of the population.
There were 120 households of which 30.0% had children under the age of 18 living with them, 60.8% were married couples living together, 6.7% had a female householder with no husband present, 0.8% had a male householder with no wife present, and 31.7% were non-families. 29.2% of all households were made up of individuals and 20% had someone living alone who was 65 years of age or older. The average household size was 2.52 and the average family size was 3.17.
The median age in the village was 38.4 years. 28.5% of residents were under the age of 18; 5.6% were between the ages of 18 and 24; 26.1% were from 25 to 44; 17.5% were from 45 to 64; and 22.2% were 65 years of age or older. The gender makeup of the village was 52.6% male and 47.4% female.