Seems I should get tired of these--but they're just so very well done, especially the acting. As we were discussing in the comments on some other post recently, most of the characters are truly believable as real people, and this is partly because they aren't Hollywood-beautiful. Frequently the women in American productions who are playing supposedly tough action-oriented characters somehow manage to convey to me the sense that they're way more concerned with how they look than with being the character.
Anyway, I have four to recommend:
Happy Valley, which really exemplifies what I was just talking about. This is not a mystery, but a cop show. The cop is a middle-aged woman, Catherine Cawood, played by Sarah Lancashire. and she is utterly convincing, as is another important character, Catherine's sister Clare (Siobhan Finneran). The valley of the title is, as you might expect, not at all happy; it's a rather depressed area of Yorkshire. There are two series of six episodes each, and each series is a complete story. My only hesitation about recommending it is that the subject matter is pretty grim. There's not much explicit violence, but there's disturbing psychological stuff.
Shetland and Vera are both based on mystery novels by Ann Cleeves. I knew nothing about Cleeves until my wife read one of the novels featuring Deputy Chief Inspector Vera Stanhope and found out there was a BBC series featuring her. Vera is a female variant of the traditional gruff-but-basically-kind-hearted cop, a rather dumpy, plain, and unfashionable middle-aged woman who is of course brilliant at solving crimes. She's a great character, wonderfully portrayed by Brenda Blethyn.
Shetland is almost as good. The biggest difference is that the detective, Detective Inspector Jimmy Perez (Douglas Henshall), is not as vivid a character as Vera. It's set in the Shetland Islands, which means you get many glimpses of great scenery. (Vera, set on the coast of Northumberland, also gives you a good bit of that.) And by the way if you wonder about "Perez", as I did: Spanish Armada.
Endeavour is the somewhat unlikely attempt to portray Inspector Morse, of the famous novels and late-'80s/early-'90s TV series, as a young man. In my opinion it works very well for the most part. Shaun Evans plays the young Morse. It's a lot to ask of a young actor to play a role in which he's expected to resemble one (John Thaw) playing an older version of the same character, but Evans is at least plausible. I notice that he uses a certain pained grimace-smile that Thaw often used. The series is set in the mid-1960s, and of course for someone my age that's interesting. Also interesting for fans of the old series are the characters who are the younger versions of characters from the old series. I especially like young Max, the pathologist (James Bradshaw). Also of interest to fans of the old series is that John Thaw's daughter, Abigail, has a small but recurring role as a journalist.
Also worthy of mention, I think, so far, is The Tunnel, although I don't like it as well as the others. It's currently being broadcast on PBS. We're about halfway through ten episodes. Like Happy Valley it's one long story, and the individual episodes are only forty minutes or so long. It's a joint Anglo-French production, featuring a beautiful and eccentric (mildly autistic, or something of that sort) young French policewoman played by Clémence Poésy. I thought she looked slightly familiar: turns out she played Fleur Delacour in one of the Harry Potter movies. It's very well done (as usual), but be warned that it is much more grim, violent, and generally disturbing than any of the others. It's based on a Swedish series called The Bridge, which I have not seen.
And be advised that I don't really care that much about the plausibility and coherence of plots in crime stories. That's definitely a fault in some of these, especially I think some of the Endeavour episodes. But it's the characters and the way their moral choices play out that keep me interested--in addition, of course, to the basic what-will-happen-next appeal of a good story.
All of these except The Tunnel and probably the current season of Endeavour are available on Netflix.