Whilst I agree in principle with robin o's sentiment - I have to disagree with his etymological attribution of Muir being anglo-saxon.

With all the usual acknowledgements:

"The Scottish surname Muir originated as name denoting someone who lived beside a moor."

and

Originally from Ireland, the Muir family's contribution to Scot's victory at the Battle of Largs secured the family seat at Rowallan Castle, near Kilmarnock in Ayrshire. Currently Clan Muir is viewed by some as a sept of the Clan Campbell and Clan Boyd.

Last I looked, my home county of Ayrshire is still located in Scotland, ergo the name is firmly of pict/celt/scots/irish origin.

Not much room for anglo-saxon derivation there.