I can’t believe I forgot the John Pyper-Ferguson connection! Pete was my second-favorite character, after Brisco himself, when I was a six-year-old kid watching Brisco in first run. Now he might be my favorite, because he’s just so weird.
I was just catching up on my missed episodes of Friday Night Lights and realized that it is not mentioned in this thread. I don’t know why, it is surely one of the best drama series of the last few years and I regularly find it heartbreakingly difficult viewing. The personal issues that the characters deal with are not turned into by the numbers TV dross. The ensemble cast are superb and the way it is shot is different to every other TV show - the actors have no blocking for their scenes, the camera operators are trained to follow the actor’s lead and it feels like a documentary.
Definitely one of the best things I have ever seen on TV. And it has sports scenes as a bonus - nothing with a sport’s theme is bad unless Adam Sandler is in it.
Another couple of votes, for Dexter and Supernatural.
I see that no-one has mentioned The Venture Bros. (probably the only Adult Swim cartoon with anything approaching a plot, sense of continuity, or general coherence ;)) or the British comedy The Mighty Boosh, which- as the Series 3 blurb says- is the TV Comedy you’d get if Salvador Dali, Dr. Seuss, and Douglas Adams wrote a sitcom after a heavy evening drinking Absinthe.
They’re both excellent shows IMHO, and well worth watching. 
I too really enjoy Burn Notice. And just for fun you might want to check out another USA Network show Psych.
I got a lot of great suggestions from this thread, though even in 9 months, I haven’t really been able to try too many of the recommendations here. However, the few I have tried have really hit the spot:
The Office (US) - The show is becoming more cartoonish nowadays, but still has one of the best large ensemble comedy casts in American television. I love little throwaways like subtle reaction shots from minor characters (Stanley and Creed in particular).
It’s Always Sunny In Philadelphia - One of the best sitcoms ever. At first, I didn’t like the show, but then I realized that they were all sociopathic assholes, with varying dashes of crazy thrown in. When I stopped caring about them as people, I started laughing a lot more. It’s essentially Seinfeld on glue, and where everyone has more than a bit of Kramer in 'em.
Six Feet Under - I’m only on the second season, but this is one of the better character dramas I’ve seen. Not super realistic in a lot of ways, but I like how pretty much all the characters are compelling and three-dimensional, and that there is not only continuity in events and facts, but in character development, too. The Mom in particular is a character that I’ve been happily surprised with; she could have been a caricature, but instead she walks the perfect line of prissy old lady and mid-life crisis gal, grasping at elusive familial and congenial bonds as if her life depended at it, and not making a bad go of it (at least so far).
After SFU, I’ll probably go for Mad Men next. In any case, thanks everyone!
My top tier recommendations:
Homicide, Life on the Streets (same creator as The Wire; I still miss it. It started Simon’s TV career of police procedurals.)
Dexter
Durham County
Veronica Mars
Bakersfield P.D. - One season of brilliance. Anyone else seen it?
30 Rock
Breaking Bad
The Middleman. Short but brilliant.
The Paper Chase