Frankenstein had "Bride", of course. A rare sequel superior to the original and a flat-out brilliant movie. "Son" is underrated. An A-list production with a great look. Karloff's last time as The Monster. "Ghost" is still pretty good. Enough new plot developments (especially with Lugosi's Igor) to keep things fresh.
Dracula had "Daughter", which is also underrated. It was way ahead of its time, with the self-loathing vampire and hints of lesbianism. "Son" was OK. Chaney didn't seem well-suited for Dracula, but it was watchable.
Invisible Man has a solid sequel with "Returns" (an early horror role for Vincent Price). Then the series starts playing things for laughs and it gets bad quick.
The Mummy movies actually weren't sequels to the Karloff movie, but a reboot. "Hand" was a fun action-horror. Then the series got progressively worse until they were MST3K-level.
The Monster-matchup movies ("House of Drac", "House of Frank", "Frank vs Wolf Man") were all fun and fast-paced, if a bit hokey.