The game is Texas Hold'em.
After the river, there are two remaining players. Player A bets, and Player B calls. Player A turns over his winning pocket Aces. Player B says "you win", and mucks his hand. Player A rakes the pot. Nobody says anything, so the next game is dealt. All is well.
Same as #1, except after B mucks, conceding the better hand to Player A, Player A says, "I'd like to see your cards anyway". Player B must show his hand (and not just to Player A; he just turns them over face up, and any third-party interested players can look to see what he had).
Same as #1, except after B mucks, somone else asks to see Player B's hand. Since the requestor was not involved in the hand (i.e. he folded earlier), Player B is not required to obey.
Same as #1, except that when Player A bets, Player B folds (instead of calling). Player A wins by default, and nobody can request to see Player B's folded hand, even Player A.
New game: Pass-the-Trash
After much betting and turning of cards, only three players are left at showdown. Two players are showing identical hands of 6-4-3-A, and Player C is showing K-K-K-J. After the declare, we find Player C is the only "high", with the other two declaring "low". Player C (as the sole "high") does not have to show his hand to anyone. He wins high by default. Player A turns over his Deuce. Player B realizes he loses low (he has a Five), and mucks his hand, telling Player A "you win". The only person who can request to see Player B's mucked hand is Player A. And if he does, everybody gets to see that Five. (Granted, both "low" players should have turned over their final card at the same time, but lets assume that Player A was eager to show his perfect low). Note that even though Player C was involved in the "final hand", he cannot rightfully request to see Player B's hand, since he was not involved in the "low" portion of the pot.
NOTE: A check on the river is considered the same as a "bet" and "call" (of zero dollars), so similar rules would apply.