I know that there is the standard bias toward the Tudors, but seeing as how Shakespeare would have had little access to genuine historical data, does he commit any serious errors from sheer ignorance?
(I feel like a charlatan. I'm not a historian, but I've got a BA in English Literature. I know what you're thinking, and yes, I am unemployed.)
The problem with such a question is that it's hard to discern 'mistakes' from the intentional deviations taken to create a better story. As with much historical fiction, there can be rather more focus on the 'fiction'.
His mistreatment of Richard the 3rd is the first thing that comes to mind. In recent decades attempts have been made to reform him in the eyes of the public, but I suspect he'll always be most well known as the hunchbacked villain of Shakespeare's play. However, this is simply a writer of fiction taking inspiration from the real world, so we shouldn't be too hard on him, perhaps. 'My kingdom for a horse' is often quoted as a genuine mistake, as horses were not readily available to people in England during the time of Richard, requiring them to be imported into the country [edit: I've been unable to find a source to verify this, so I'm retracting it]. But again, as to whether this is a mistake on old Bill's part or whether he just didn't care, it's difficult to know - I'd be half tempted to say we can't know, but I lack the authority, so I'll leave it for someone else to say.
It's important to know that Shakespeare's Richard didn't exist in a vacuum or come from nothing - sources painting him as the ultimate villain pre-date Shakespeare. Richard's contemporaries were quick to fall in with the anti-Richard rhetoric that immediately followed his death, and these were just the first of many in the subsequent generations. So Shakespeare's inaccurate depiction of Richard was likely inspired by these deeply biased sources from invested parties dating back generations.
In Macbeth we get more of the same, though his inaccurate depiction of the historical Scottish king has garnered less fevered attempts at reform in the public eye - perhaps because there is even less known for sure about the historical Macbeth than there is of the historical Richard. The name 'Macbeth' will, for as long as there are still people around to speak the name, be associated with a diabolical madness and ruthless acts of barbarism. As far as I am aware, that's entirely Shakespeare's doing, as so little is known of the historical Macbeth; I can't imagine he (Shakespeare) would have had access to any significant sources on the man, and I struggle even more to believe that they would contain information enough to prove useful to him, and result in the Macbeth we know and love to hate. I think Shakespeare alone must bear this particular cross.
Sources:
Fiction:
Non-Fiction