For the English, you must prove that you have a direct line of blood through male heirs. That means most of you can't claim your family COA. (I'm a herald, I've done research, even if it was a while ago) My family's name is Brownlow, and if you're in the UK, you've probably heard of it. There was a baron Brownlow, and there is a Brownlow Manor.
I've been able to find several different versions of the Brownlow coat of arms. One of the Brownlow family was given desmene over land in Ireland, so there's a seperate COA for that branch of the family (it never went anywhere, everyone was killed). For reasons I haven't explored there is no longer a Baron Brownlow in the UK, so the arms are now mostly seen around the Manor and in a few pubs. My Family arms are pretty simple, per pale or et argent (divided in half, gold and silver(or yellow and white)), an escutchean within an orle of martlets sable (a smaller shield in the middle of the primary shield, and a ring of birds around it, the small shield and birds all black).
Now, the coat of arms I'm using as my avatar is my SCA coat of arms, I designed it myself using the proper conventions and registered it with the society college of heralds. It is unique in our society (and probably even unique in the world). We do not allow certain things in your arms if you haven't earned it (that is to say that certain symbols are restricted), and any official royal heraldry for the various kingdoms and families is also restricted. We have literally thousands of COA's registered, and they all are different and follow the rules of heraldry. Plus, you only get to use them once you have earned your Award of Arms (same thing as in real life, that's how individuals today get new COAs).
Hope that has educated some of you, and if you have any questions or perhaps want help creating a COA for your own CC faction, ask me (Or Freak, he's pretty good at it to). I've also got TONS of heraldic clip art that I use in the SCA, so if you're looking for something in specific, request away!