There appears to be nothing in the Bible that gives a real clue as to what the dispute between Michael and the devil could have been about, further to Jude verse 9. They had a dispute about Moses' body, which ended with Michael calling on God to rebuke the devil. We do not know any details about the actual dispute.
All we may safely claim is that the devil was saying nasty, untrue things about the meekest of men - Moses. He was speaking ill of the dead. The previous verse in Jude refers to wicked men who "defile the flesh, despise dominion, and speak evil of dignities [Lit. 'glories']." Then the devil is mentioned, perhaps as the worst example of such wickedness. The Bible commentary below ventures to suggest this:
"This dispute must have taken place after the death of Moses and his
burial by Jehovah, for "death reigned from Adam to (until) Moses"
(Rom. 5:14). The devil claimed Moses for the death-state, but God
raised him as representative of those hereafter to be raised, as
Elijah of those to be caught up without dying." The Companion Bible
Bullinger, page 1882, side-column re. verse 9
There is, however, a Jewish tradition, "Assumption of Moses" (or, The Testament of Moses), a 1st century apocryphal work known only through a single 6th century Latin text in the Biblioteca Ambrosiana in Milan. It claims to have secret prophecies given to Joshua by Moses before he died, but very little of that document exists. See: What is the Assumption of Moses? (Got Questions)