How common was it? was it more common among the poor? or does the stereotype of an inbred English member of the Aristocracy got some truth to it? Was there a stigma to it that was ignored for the sake of the political benefits marriages can give or was it considered acceptable? Or was it not really considered an issue at all?
I'm really just genuinely curious when I realized this is something I've never read a single thing on but may be very significant and worthy of a read.
Thank You.
Genealogical records of the poor are scant (sources below). It is landowners and nobility that are represented in civil and church records, particularly during the period you mention (during Elizabeth times, a rising middle class is more and more represented in church records: millers, brewers, pubowners, scholars, teachers, tradesmen).
So, we don't know a lot about cousin marriages among the lower classes. But, it is safe to say that poor people would have had even fewer resources to get outside local villages/neighborhoods to find partners. I'm guessing that trends in Britain parallel similar social conditions in other societies where data is available, in that urbanization leads to less inbreeding. Since many Brits were still rural in the period you're asking about, many would have married locally (hence increasing the chance of cousin marriage, particular 3-6th cousin marriage).
As for the upper classes, attempts were made to avoid cousin marriage, the Church prohibited marriage within 7 degrees, records of dispensations though show that it was still common enough (and in my own tabulations, higher status individuals were more likely to have a shared ancestor within 3-4 generations on both sides of their pedigree).
The ebook by the New York Public Library (below) easily illustrates this. Most of the records involve such things as applying for a coat of arms or official pedigrees (created by specialists and paid for by families who were trying to maintain their connections to the aristocracy). Heraldry, coats of arms, and various lists of the peerage are main sources for British genealogical information, especially in the middle ages and early modern periods.
Cole, Jean A. and Rosemary Church. In and Around Record Repositories in Great Britain and Ireland. 3rd ed. Ramsey, Huntingdon, Cambs: Family Tree Magazine, 1992.
Colwell, Stella. Family Roots: Discovering the Past in the Public Record Office. Rutland, Vt.: C.E. Tuttle Co., c1991. 231 p.: ill.
Eakle, Arlene H., Arvilla Outsen and Richard S. Tompson. Descriptive Inventory of the English Collection. Salt Lake City: University of Utah Press, 1979. xvi, 168 p.
Moulton, Joy Wade. Genealogical Resources in English Repositories. Columbus, Ohio: Hampton House, 1988.
List of Works Relating to British Genealogy and Local History. New York Public Library (freely available as a Google eBook).