The most memorable criminal case I ever stumbled upon in a video game is Arcanum's Half-Ogre Conspiracy quest.
The best (or worst, depending on who you ask) part about this case is that technically you cannot solve it.
It's a vast, X-Files-type government conspiracy that presents a stark contrast to the rest of the game. Sure, Arcanum was always about race relations and exploring similar urban (or steampunk, I guess) fantasy tropes, but these usually played second fiddle to classic CRPG mechanics and power fantasies such as having your actions shape the destinies of entire countries.
That is, until the Half-Ogre Conspiracy. Here, you learn a very uncommon videogame lesson. Namely that, no matter how powerful or clever you are, there are some problems that you cannot solve, regardless of how hard you try.
The entire case also took a much darker turn than I originally expected, dealing with some very disconcerting themes (which I will not spoil here), but is still well-hidden enough that most players have very little chance of stumbling upon it (though its existence is hinted throughout the game).
But even if we disregard all of these themes and meta-narrative implications, it's still a great case - leading you from one revelation to the next. The fact that it is so thought-provoking and different than anything else in the game is just an added bonus.