I don’t have any contributions, because all the mystery series I can think of were either canceled without answers, or dragged on and on until a disappointing end.
The only mystery show I can think of watching that left me feeling ‘satisfied’ usually closed with the words “and I’d have gotten away with it too, if it weren’t for you meddling kids!”
I liked the ending to MONK.
I’m not exactly sure this fits in with what you were looking for, but the first thing I thought of was MAS*H.
Hawkeye has a mental break down, and the episode unravels a mystery (so to speak) where throughout the episode the gang was trapped in a bus in enemy territory, and a woman’s chicken was making noise, and they were afraid they were going to get caught. The mystery of course is that the woman suffocated her baby to save everyone’s life. I think I’d have a little anxiety over seeing that as well. 
Yeah, that’s right no spoilers! It was like 30 freaking years ago!
The terrific Granada TV series of the Sherlock Holmes tales just ended with Holmes and Watson living to solve another case on another day… and that’s just how I wanted it to end.
Yeah, I can see now that the OP wasn’t too clear. I was thinking more about shows with a story-line that contains a major mystery that isn’t solved for at least one American* TV season. Even if solving the mystery isn’t the major focus of every episode. Like the disappearance of Mulder’s sister, or “Who killed Trudy Monk?” I forgot about that one and agree that it was good.
*Doesn’t have to be an American show, but it wouldn’t be fair to compare a five episode long British season to a 22 episode long U.S. season.
I nominate Veronica Mars. Especially the first season.
IIRC, all three seasons of the Australian show “Miss Fisher’s Murder Mysteries” have an overall mystery to solve along with the mystery-of-the-week. All handled quite well. The first two are 13 episodes each but the third has only eight.
For shows that had an over-arching mystery at their core, such as Twin Peaks, John Doe, X-Files, I can’t think of a single damn one that had a decent resolution.
For shows like Monk that had an underarching mystery, well, Monk is the only one I can think of. Definitely not Burn Notice.
The UK program “New Tricks” had a multi-season underarching mystery regarding the death of a main character’s wife. Satisfactorily resolved, AFAIK.
The Fugitive, of course.
Ooo, good one.
The comedy Soap started with a story-arc mystery (“Who killed Peter Campbell?”) and it occurs to me that the genre the show was spoofing - daytime soap operas - also had multi-episode mystery arcs. I’m sure some of them had satisfactory resolutions.
The UK series “life on Mars” and “Ashes to Ashes” had a really neat (if not entirely comfortable) resolution.
I’m going with Wayward Pines, season 1.
So, I’m not saying that the “answers” were particularly smart or realistic or that they stood a second viewing… but there was a lot of weird stuff going on, like really weird, and then an attempt at giving an explanation that made some sense.
You know, as opposed to some shows that set up a lot of questions that go either unanswered or are waved away as *“The island is just magic and everything that happened was a game between two demigods consisting on just doing a lot of random nonsense just for kicks”. *
Key: The Metal Idol
And the reason that I continue to argue that TV series need to decide an ideal number of episodes for a series, learn how to write for it, and stick to it. If a series does well, then they should come up with a new story - maybe even a new set of characters - rather than just accepting that there’s going to be more episodes each year.
Broadchurch, the first season.
While Fringe perhaps wasn’t perfect in its final season, it did supply an answer to its central mystery, and the ending hung together quite well.
That’s what I thought of. And season 1 of Murder One.
Bingo.