The most memorable criminal case I solved in a video game was bringing the Death Angel to justice in the original Police Quest, where I went undercover and had to win at poker in order to get close to him.

For me that would be the investigation and apprehension of Jessie Bains (aka The Death Angel), in Police Quest. What started as a "regular" police investigation became a rollercoaster of breakthrough and fear, as Bains grew from a street thug to criminal mastermind and later to a desperate criminal seeking reckless personal vendetta against the protagonist (later continued by his brother).

An amazing series of games, amid the golden age of adventure point-and-clicks / text-parsers, that pushed storytelling to a level that still holds today.

Larry Laffer and his all criminal stories...

Cognition : You play as Erica Reed, a FBI agent who has the ability to see into the past of any object that she touches.
Haunted by the murder of her brother by a crazy serial killer, whose case has been closed, you will then try to solve a serie of murders which, as the game progresses, seems to get you a little closer to your brother's murderer.

Playing the game of cat and mouse with the serial killer was really compelling; the twist at the end was really mind boggling.

The game has some flaws, but these are quickly forshadowed by its strong story.
Besides, it's always a pleasure to support indie games developpers.

I remember the missing painter case in Oblivion. It was one of the most original side quests in the game with a surprising development...

Solving the disappearance/murder of Rachel Amber in Life is Strange! Digging up Rachel's body with Chloe was so emotional. Then being kidnapped and coming face-to-face with her murderer was such a crazy way to find out.

Ultima VII. The murder at the stables.

Probably the first (unaltered version of) Broken Sword.

Very few games can hook the player from the very first moments, and keep the flow of the game relatively good throughout the experience.

For me the most memorable criminal case in a video game would have to be the Shanghai Justice quest from Deus Ex: Human Revolution.

Adam Jensen's pilot Faridah Malik had a very good friend back when she lived in Hengsha called Evelyn Carmichael. Evelyn was in a relationship with Lee Hong, son of a very powerful and influential Chinese investment family. Evelyn loved Lee, but he was very arrogant and didn't want to commit to creating a family with her not to mention that his family didn't approve of her.

One day Evelyn died and her death was ruled an accident. Faridah however doesn't believe it was an accident and wants Jensen to help her investigate the case again.

From start to finish it is a thrilling narrative and one particular way how the quest can end is pretty epic.

Also, according to the Deus Ex Wiki, the dialogue in the quest apparently references Christopher Nolan's movie Insomnia (2002) which also centered on a criminal case involving murder.

I think LA Noire just because it actually allowed you to interrogate suspects along with searching for clues. It may not have been the most well-implemented interrogation system, but it was quite an interesting mechanic and the facial expressions were utilized well in that aspect.

In all honesty I don't remember much about the murder case I solved in EAs

The Lost Files of Sherlock Homes: The case of the serrated Scalpel

but I have vivid memories of the faboulus graphics and the all over mood transporting the victorian era athmosphere right into my parents office.rather excellently.

Solve the mystery of The Obra Dinn, a great game!

Having played many detective games on gog such as Batman games, Deus Ex games and Vanishing of Ethan Carter, I have to say that the most memorable criminal case solved in a video game for me would have to be in the game called "The Shivah".

Without spoiling anything, what struck me the most was the fact that the person who committed the murder was from the same background as you and very well known almost like a family.

Life lesson learnt - Money is far more powerful of a force than religion.

Solving the murder mystery in skyrim!

Without a doubt it was The Secrets of Loc Muinne, from Witcher 2. The assets that were created for this quest alone were mind-blowing! The writing was also a cut above even Witcher 2's standard and brings to mind the mysteries of the Witcher world at large. The Talk Golem was a fierce duelist of words, and could not have been more unique in that era of games of any genre. Dearhenna, a character entirely absentee, also proved most intriguing. His discovery of the [No Spoilers!], whoa!

I'd call it, with full confidence, the single most memorable quest of any game I've played, and the reason for why I have such a high opinion of Witcher 2.