Certainly a contentious subject. However, it has been firmly established that hereditary 'arms' were first used on 9th century seals. From there they graduated to being displayed on lance pennons and it is these pennons that are depicted on the Bayeux Tapestry. Most, if not all, of the lance pennons with 'heraldic designs' are carried by the Flemish contingent of duke William's invading army.
The subject is more fully examined in the book "Origns of Heraldry" (ISBN 0-906650-00-3) by Beryl Platts (1980) which traces the Flemish families involved with the Battle of Hastings. She subsequently produced two more books, "Scottish Hazard" in two volumes; Vol.1 - 'The Flemish Nobility and their Impact on Scotland' and Vol.2 - 'The Flemish Heritage', both of which further explain just how the arms of the Flemish aristocracy managed to appear in the earliest rolls of Scottish arms. Apparently there was quite a large exodus post-1066 from Flanders to Scotland and several Scottish families are of distinctly Flemish orign; the Lindsays who stem from the Flemish Alost family who appeared in the Bayeux Tapestry in the guise of Baldwin of Alost and his brother Gilbert of Ghent. Gilbert was the father of Walter de Lindsey, the founder of the Lindsay family name in Scotland.
Another Flemish family represented in the Bayeux Tapestry is that of Bethune (Hereditary Advocates of Boulogne) who founded the Scottish Beaton family. The arms of the two family branches only differed in the colour of the field, the Gules field in the Bethune arms being replaced with the Azure field in the Scottish Beaton arms.
There are another couple of lance pennons that are depicted in the Bayeux Tapestry that have been firmly established as eventually appearing in both English and Scottish early rolls of arms.