What exactly is the difference between these words. When do you use one, and when the other. Dictionaries aren't too helpful, as they list them as synonyms.
Insignia: A distinguishing badge or emblem of military rank, office, or membership of an organization.
Crest: A distinctive device representing a family or corporate body, borne above the shield of a coat of arms (originally as worn on a helmet) or separately reproduced, for example on writing paper.
Emblem: A heraldic device or symbolic object as a distinctive badge of a nation, organization, or family.
Badge: A small piece of metal, plastic, or cloth bearing a design or words, typically worn to identify a person or to indicate membership of an organization or support for a cause.
A badge is typically thought of as the as the physical object that that has an emblem or crest printed or embroidered upon it.
Whilst the above dictionary entries have overlaps, insignia is most commonly used in relation to the military, crest most commonly used in relation to family and corporations and emblem is more nationalistic.
A badge is originally, and still is; in heraldic terms, any emblem used by an armiger (a person with an achievement of arms) on his standard (a type of flag which is not a banner) and used by his followers.
For instance, the three feathers in a crown, with the text "Ich dien" (I serve) under, is the badge of the prince of Wales. And as a badge, it is not used by the prince himself, but rather by his staff, on letter paper and so on.