Frankie Drake is the first that comes to mind. Agent Carter would also fit your criteria, although it's a very different genre.
Movies and TV Shows
General discussion about movies and TV shows.
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the crazy movie ending that no one saw coming!
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Carnivale, Boardwalk Empire, the last season or two of Downton Abbey, Call the Midwife, Marvelous Mrs Maisel. All of those came up on search (amongst many more), and I enjoyed them. Mrs Maisel is one of my favorite shows ever.
I Love Lucy started in '51. Les Paul & Mary Ford at Home started in '54 (shorts, music & skits). The Honeymooners started in '56. The Donna Reed Show started in '58. Dobie Gillis started in '59.
Bear in mind that TV shows in that period usually displayed a highly idealized way of life (The Honeymooners less than the others). If you want to know what life was like in that period, Leave it to Beaver isn't going to help you. Books are your friend there.
Babylon Berlin might be right up your street!
The Untouchables has both options for you.
Movie being the most famous.
But there was also a tv series from 59-63 and a reboot 93-94.
Movie, not a show but Miller's Crossing is a masterpiece
lmao LA Confidential cast a shadow underneath Miller's Crossing
A Private Affair, Spanish PI show with Jean Reno as the trusty sidekick