The 'correct' (i.e., original) singular form of insignia. If one is quite picky about one's grammar, one would use 'insignia' only when referring to more than one insigne. This usage comes to us directly from the original Latin (also the root of our word 'ensign'), and was still used in common parlance through the 1700s.
However, this use of the original singular form pretty much died out by the late 1800s, and 'insignia' has come to be used as both singular and plural; to avoid confusion, the plural form 'insignias' has started to come into popular usage. This is still considered incorrect by many, and one can gain a fairly good measure of a pedant's pedantry by finding where they stand on the insignia issue.