Lower, Mark A (1860) Patronymica Britannica: a dictionary of the family names of the United Kingdom. London: J.R. Smith. Public Domain.
We don't have any stats on how common this name is. This is probably because it's very rare in the UK.
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The Mure surname in historical dictionaries
Patronymica Britannica (1860)
MURE. 1. The northern form of Moor. 2. Gael, viohr or more. Large in person. The family of Mure of Caldwell in Ayrshire trace to the reign of King David II., 1329. The name has been 'aried to Blore, JIuir, Moor, (fee, and there are hei'aldric reasons for believing the Moores of Kent and those of Ireland to have had a common origin.