I realise they'd be variations of deployment depending on the numbers, terrain, nature of the foe etc, but generally speaking how many men deep would they go?
Also more specifically, I have read that reliable sources for Stamford Bridge are scarce, but were there any reports at Hastings on how deep the line went for these battles? Any more information on shield wall tactical variations and general deployment would be interesting to hear too.
The short answer is that no, there are really no detailed tactical reports for pretty much any early medieval battles. The Anglo-Saxon Chronicle D claims William attacked Harold 'before his [Harold's] army was drawn up in battle array', so in that particular case it may be that Harold's army had not yet managed to form up into a proper shield wall. As far as how deep, the closest suggestion from the sources is from the thirteenth century Saga of Hakon the Broad-shouldered, part of the 'Heimskringla' sagas of Snorri Sturluson, which mentions a shield wall that was 'long and no more than five men deep'. Whether this has any real basis in actual early medieval shield walls is debateable.
Sources: Douglas, D. C. and Greenaway, G. W. (eds.), English Historical Documents Volume II 1042-1189 (London, 1953), p. 144
Halsall, G., Warfare and Society in the Barbarian West, 450-900 (Abingdon, 2003), p. 195