Are there any popular, well-known anime series that share characteristics with the film noir or neo-noir style or setting?
Film noir was originally a term used by French critics to describe Hollywood movies of the post-war 1940’s era (up through the 1950s) that portray a darker perspective upon life than was standard in classical Hollywood films and concentrate upon human depravity, failure, and despair. After 1960, films of similar nature that actively try to emulate the classic noir style (for better or worse) are deemed neo-noir. Both terms typicall imply use of a cinematic style with a particular way of lighting, positioning and moving the camera, and using retrospective voice-over narration
There are no anime series that can be considered truly film noir in genre as they typically use the style but not the motifs and cinematography, identified by the genre.
I believe the following series most closely resemble the noir/neo-noir-style style, story, and/or setting (some not completely) the following would probably fit:
Both Big O and Cowboy Bebop have characters being plagued by events in their past and fatalistically having to accept them. Big O uses the style of film noir a lot more and uses its premise as the basis of entangled plot. Cowboy Bebop, on the other hand, simply sticks to the premise and borrows its style from elsewhere.
Baccano! is like a blend of film noir and pulp fiction, where a bunch of seemingly unconnected stories slowly become connect into one overarching storyline, which is very typical of the pulp noir genre.
Ghost in a Shell and it's sequel are considered a marriage of noir visual, sci-fi, and animation, and sometimes included under the subgenre of noir called tech-noir, which includes movies such as Brazil and Blade Runner. "A Detective Story" from the Animatrix is considered to be another example of the above mentioned tech-noir genre.
Perfect Blue has the mystery setting, murder cases with no apparent suspect and a main character is being hunted, keeping the story on the edge. It also deals with amnesia, flashbacks, uncertainties of one's own memories.
Bodyguard of the Wind Though not as well -known, has all the motifs of a classic noir detective story. It opens with the disappearance of a train carrying a shipment gold along with its guards, but the story really starts with the son of one of those guards coming to the city where the train vanished to investigate. There he is met with the residents many of hiding their own secrets and mysteries, forcing him to slowly progress from one clue to the next, like a noir detective would do, to get to the bottom of everything.
These don't really fit the noir-style completely for one reason or another, but are heavily influenced by the noir genre:
The Skullman has a noir-ish atmosphere is given by the mystery which shrouds the strange murders in that alternate history of the end of WWII.
Speed Grapher has the noir atmosphere and art, but the story and characters aren't what you typically expect of the genre
Texhnolyze has plenty of elements from "noir" movies: dark lighting, flashbacks that disrupt the normal flow of the story, a corrupted society, a sort of "femme fatale" and a main character that seems to not fit the rest of the world. However, this is more of a sci-fi/cyberpunk anime with a dystopian setting.
Daughter of Twenty Faces is more of an action comedy that uses Edogwawa Rampo (a well-known Japanese noir writer) characters and the noir-style of mystery behind it's plot.
Are there any anime that would be consider either Noir or neo Noir?
Dictionary definition of film noir would be: a film marked by a mood of pessimism, fatalism, and menace.
Stereotypical elements are: murder, a detective who drinks whisky late at night in a shabby bar, and a femme fatale.
Anime that I have seen and that more or less match above description are: