Treatment of a Grignard reagent with oxygen gives the magnesium organoperoxide. Hydrolysis of this material yields hydroperoxides or alcohol.  These reactions involve radical intermediates.
The simple oxidation of Grignard reagents to give alcohols is of little practical importance as yields are generally poor. In contrast, two-step sequence via a borane (vide supra) that is subsequently oxidized to the alcohol with hydrogen peroxide is of synthetic utility.
The synthetic utility of Grignard oxidations can be increased by a reaction of Grignard reagents with oxygen in presence of an alkene to an ethylene extended alcohol.  This modification requires aryl or vinyl Grignards. Adding just the Grignard and the alkene does not result in a reaction demonstrating that the presence of oxygen is essential. The only drawback is the requirement of at least two equivalents of Grignard although this can partly be circumvented by the use of a dual Grignard system with a cheap reducing Grignard such as n-butylmagnesium bromide.