Impossible checkmates

I'd like to see checkmates positions which are impossible to achieve in game.

The last position (wKe1, wRg7, wRh7, bKh8) is in fact possible. Last move was Rxh7, capturing anything except a pawn. Something like this is illegal though:

@Remellion: The last position too is impossible.

@ Remellion: you are right about the last position, if Black cooperates it is possible.

As for your example, I don't understand why it is illegal. Is it because the pawns are forced to move at times?

Anyway, more examples:

If you try to reach the position I gave from a legal game, you'll never be able to. To see that, try asking "what was black's move before white's last move?" - that may turn up some issues.

This too is illegal. It's actually not as easy as you might think to show that a position is legal or illegal - some make good puzzles.