Buckingham was executed in Salisbury on November 2, 1483, after his repeated requests to see Richard III failed. He may well have made a will before his execution, for both Thomas Catesby in his pre-execution will and an act of Henry VII's Parliament restoring his widow her jointure make reference to a will by Buckingham. Probably, like Catesby and Anthony Woodville, he requested a final resting place. Unfortunately, Buckingham's will, if he had one, does not appear to have survived.
The first candidate put forth as a resting place for Buckingham was the Church of St. Peter in Britford, just outside of Salisbury. In 1836 in The Gentleman's Magazine, R. Colt Hoare wrote that the tomb bore the shield of Stafford and Rivers. He added,
I am inclined to suppose that the figures on the base of the tomb allude to a melancholy event which took place at Salisbury. There are six niches, five of which contain male and female figures; the first is vacant, which I think was designed for the unfortunate Duke. I consider the female figure in the second niche, having a crown on her head, as representing the Duchess, his wife. The next figure is evidently an ecclesiastic or bishop deploring the unfortunate fate of the Duke; and at this period Widvile, brother of the Duchess, was bishop of the see. The fourth figure represents a female crowned like the second, holding a sword in one hand, and in the other a cap or bonnet, probably that of the Duke.
The fifth figure represents the executioner with the sword in his hand.
The last figure represent [sic] a female holding up her hand in apparent grief, and with a child in her arms, as alluding to one of the unfortunate Duke's offspring.
One D.H., however, also writing in The Gentleman's Magazine, would have none of this. Dissenting in a gentlemanly manner (as one might expect), he wrote that the figures probably represented saints, not family members of Buckingham, and he doubted that the shields represented the House of Stafford. Later writers have suggested that the Britford tomb might have been erected in Buckingham's memory, but does not contain his remains.
The picture grew murkier in 1838, when according to a report in the Salopian Journal, during renovations at Salisbury's Saracen's Head Inn, a skeleton was found beneath the flooring, missing its head and right arm. The skeleton underwent an extremely unscientific examination by the locals, with the landlord measuring a rib against his own and a maidservant "laying irreverent hands upon the neck-bones." Under the assumption that the skeleton belonged to a long-ago murder victim, the workers knocked the fragile bones about so that they merged into the surrounding clay. A few 19th-century antiquarians suggested that this was Buckingham's skeleton; they noted that the Saracen's Head Inn stood on the site of the Blue Boar Inn, the yard of which is given by some sources as the site for Buckingham's execution. It's possible that Richard III, furious at Buckingham's betrayal, might have ordered that his erstwhile ally be buried ignominiously instead of in consecrated ground, but how to explain the missing right arm?
Finally, the Chronicle of the Grey Friars of London states, "Thys yere the duke of Buckyngham was be-heddyd at Salsbery, and is burryd at the Gray freres [in Salisbury]." As noted in a footnote by J.G. Nichols, this is probably the most logical resting place for Buckingham. It was nearby, and as Richard III had afforded other executed opponents of his, notably William Hastings, honorable burial, he probably did so with Buckingham as well.
Wherever the duke lies, he does not seem to be doing it quietly. Buckingham's ghost is said to haunt Debenhams department store in Salisbury, where he has been known to enter the women's dressing room and hiss, "It makes you look fat, darling." (OK, I made up the last line. But not the haunting Debenhams part.) The site of his execution is marked by a plaque on the store.
So if you're shopping at Debenhams and encounter the ghostly Duke, why not gently ask him to clarify the matter of his burial? Oh, and ask him about the Princes too while you're at it.