What I have seen are:

- Illegal moves, particularly with knights.

- Adjusting pieces onto squares they are not on: edging a rook towards another square slowly, then "adjusting" it onto that square to essentially get a free move.

- Dropping pieces and putting them on new squares.

- Double capturing: picking up your piece as though they are going to take it with theirs, then moving one of their pieces to take a different piece.

This summarizes it well. Two more (in fact, the most frequent ones I saw) are:

(1) He moves a piece, accidentally another piece tumbles down on the board, he adjusts it, and meanwhile, one of the opponent's crucial pawns magically disappear.

(2) Same as above, with opponent's pawn (or his pawn) magically moving to a better location for him.

I have played twice with them and they made illegal moves on one and the time ran out on another. I cannot confirm if my opponent was using hacked clock (I strongly believe he was) but I am pretty sure his diagonal move was illegal. He argued his piece was somewhere else and I got fed up and left.

You can see a classic example here (around 2:20): https://www.youtube.com/watch?v=U5vnpOp0U_g