...interesting topic.
The tough thing is trying to mention titles that fall under the "police/detective/murder" sub-category of the adventure genre. This may prove to be hard, since sometimes "fantasy" or even "horror" adventures are coupled with detective-work, and mystery solving in general.
As such, I tried to leave out titles like "Shadow of the Comet" that are clearly leaning more on the "horror" side, than the "police/detective/murder" one.
I'm going to start off with Beam Software's
"Sherlock" as an honorary mention, since it's the first detective-story adventure I played, back when my english sucked big-time (yes, even suckier than my present use of english

).
Speaking of Sherlock Holmes, you can also try the (old-but-good):
"Lamberley Mystery",
"The Case Of The Beheaded Smuggler", and then move on to the more modern
"Lost Files of Sherlock Holmes" series (Serrated Scalpel & Rose Tattoo), and up to the present with the recently released
"Secret of the Silver Earring".
Sierra's
"Police Quest" series (not just the 4th one) also fill the requirements nicely.
Another enjoyable "detective-work" mystery-solving series is "Nancy Drew" - the most recent one being the
"Secret of the Old Clock", IIRC.
Yet another memorable series is the "Tex Murphy" ones. Starting out with the classic
"Mean Streets" and
"Martian Memorandum" and moving on to the FMV-era sequels, like
"Under a Killing Moon",
"Pandora Directive", and
"Overseer" (that last one was my first DVD game, IIRC).
...and while we are all over the FMV era, why not try such titles as
"Psychic Detective",
"Ripper", and the
"Black Dahlia", or the more "cartoony" (but equally atmospheric)
"Jack Orlando" &
"Private Eye" (which features Philip Marlowe in a very well known story/case).
Now moving back to classics again.
Since "The Dagger of Amon-Ra" was mentioned, make sure you try the excellent prequel
"The Colonel's Bequest".
"A Cruise for a Corpse" is another police/detective story I can't reccomend highly enough.
Some more treasured oldies of mine are:
"Borrowed Time",
"The Case of the Mandarin Murder" (featuring Perry Mason),
"Scapeghost" (SUPERB game), and
"Corruption" (simply brilliant).
When
"Deja Vu" was released by the "Macventure team" (later to be known as "icom simulations") it was an instant hit on the Mac, and found its way to other popular computers as well. Make sure you play it, along with its sequel
"Deja Vu 2 - Lost in Las Vegas".
If you are more of a console-junkie, you can also try the simplistic (like all console adventures) but still very entertaining detective stories:
"Famicom Detective Club 2" and
"Snatcher".
...I'll try to remember some more adventures that fall under the "detective/police/murder" category if I can...