Thriller (1960-1962). This Boris Karloff hosted (and sometimes featuring) anthology show started off with more standard fare about murders and blackmail plots and the like, but soon veered into much darker territory with stories about ghosts, vampires, and not a few Edgar Allen Poe moments. This is one that I discovered on MeTV on Sunday nights, and it's well worth setting your DVR for. Standout episodes include "Masquerade" (with Tom Posten and Elizabeth Montgomery in a fantastic episode with a lot of humor and a great twist ending), "Flowers of Evil", and "The Remarkable Mrs. Hawk".
The Outer Limits (1963-1965). This show was an obvious effort to capitalize on the success of its close cousin, the Twilight Zone, and overall it's frankly not as good, but its best episodes stand up against the best of its erstwhile competitor. Standout episodes include "The Sixth Finger", "The Zanti Misfits", and "Soldier".
The Night Gallery (1970-1973). The follow-up to Rod Serling's Twilight Zone, the Night Gallery skewed much more towards gothic horror and featured more ghosts, psychics, and the like. It was also not above showing more humor, although the gimmick of the art gallery doesn't really quite work. Stand out episodes include "The Boy who Predicted Earthquakes", "Marmalade Wine", and "Little Girl Lost".
Kolchak the Night Stalker (1974-1975). This show starring Darren McGavin as reporter Carl Kolchak, a Chicago reporter who keeps stumbling on stories involving the supernatural, was just fabulous, and died before its time. Darren McGavin's rumpled reporter was played wonderfully broadly, with lots of humor to break the tension. Long suffering editor Mr. Vincenzo was another stand-out character, who just wanted Kolchak to report on normal stories. There were also two made for TV movies; The Night Stalker and The Nigth Strangler. Stand out episodes include "The Energy Eater", "Firefall" (featuring a pyromaniacal doppelganger), and "The Zombie".