I am trying to find the most common way to refer to a soccer team's shield (or coat of arms).
Checking the websites and wikipedia pages of half a dozen football teams, badge seems to be the most popular term, followed by crest and logo.
This seems like a question for which Google Ngrams should be perfect; but oddly, looking for phrases like
Arsenal badge there, I haven’t managed to find any hits for any combination, although Googling shows that they are used plenty often in the wild. I suppose there just can’t have been many sports texts in the corpora that Ngrams is based on.
Perhaps logo (an identifying symbol, often for use in advertising) or insignia (a distinguishing mark or sign)...?
I've got a towel with Manchester U's logo on it, because I love being able to wipe the floor with them.
Edit to add: I would guess that "logo" is the most commonly used term since it is advertising-oriented and most of the time I'd expect that their mark is being referred to in an advertising-related way (selling towels or shirts or other marketing materials, for example).