Only certain daily dramas. I take a look at the storyline first--it has to interest me. Once in a while I watch it because one of my favorite actors/actresses is the lead. But if I see that the drama is heading into nonsense, I usually drop it.
For the most part, I like watching KBS TV Novels. They are morning daily dramas and the time period is usually in the 1960's, 70's, or 80's. The love relationships feel a bit more simple in nature (at least to me) when compared to the mini-dramas. A plus is that the TV Novels are similar in vein to ATV's The Good Old Days, TVB's Plain Love/Plain Love II. They may or may not involve a birth secret.
My favorite TV Novel so far is "You are a Star"/"My Star, My Love" from 2004. "Wind Flower", which followed "You are a Star" was also very good. Though I liked watching "Dear My Sister", which aired in late 2011 to May 2012, there were way too many annoying (that got on my nerves) characters. And unlike the older KBS novels, didn't apply an outside narrative voice. "Love, My Love"/"Dear Love" (currently airing) has been good so far. No real annoying character, unless you count the Grand Uncle who is leeching off of his older sister's family. Will have to see whether 2 half-sisters will fall for the same guy (doesn't look like it so far) and whether 1 half-sister will fall in love w/ her unknown step-brother. Hehe, sounds complicated already, right?
As for general daily dramas (those taking place in the current time frame), I actually don't watch many and/or there isn't many that I like. My favorite is "Be Strong, Geum-soon" and maybe follow that with "Yellow Handkerchief" (the daily drama that got me into K-drama). I don't think I have watched or followed through with daily dramas after 2006. However, I am watching "I'll Give You the Stars and the Moon"/"Moon and Stars for You" because I like actress Seo Ji Hye (love her role in the saeguk Shindon). Going to be interesting when the ambitious girl (not Song Ji Hye) finds out the rich guy she's currently pursuing is her nemesis' (Seo Ji Hye, from high school days) brother (adopted), that the doctor guy she dumped will have a romantic relationship w/ her nemesis, and that the rich guy might be the doctor guy's brother or related family member?! I just hope that the rich guy doesn't turn baddy because his adopted mom sees him as a threat to her daughter (Song Ji Hye) in taking over the family business, thus not treating him well.