Sandra Clark as portrayed by Jackée Harry on 227 is the one answer to this question that I will never forget, even as I forget most of the specifics about that show. I don’t want to kick the actress, she was given a role, developed a character in conjunction with the director … but, damn, that annoying “Maaaawwie”, ugh, why, just why, was that considered entertaining.
Some would add Steve Urkel, I wouldn’t, I think he took annoying and made it charming. But I might be too generous, and I didn’t really watch that show.
Megan Mullally as Karen Walker on *Will and Grace * might also fit the bill. I only saw it for a few moments, but I got a quick feeling that this was another Jackée Harry-esque character. Again, actress was doing what she was supposed to do, I guess, but who decided to write a show around annoying?
Fawlty Towers is a tough one. Basil is an ass, but Sybyl doesn’t actually work, Manuel is pleasant, but pretty incompetent. And Polly isn’t really the voice of reason. I guess it is an entire show written around being annoying. So it can work.