I'm very, very surprised by the absence of "The League of gentlemen" in the lists suggested above.
That was one that kicked off the sofa. Actually, I'm just looking for the DVD box in the net shops.

And what about On The Busses and Doctor In The House? There's just so many. I hope one day that Netflix has an entire grouping of these great shows.

Kalle Rantaaho
I'm very, very surprised by the absence of "The League of gentlemen" in the lists suggested above.
That was one that kicked off the sofa. Actually, I'm just looking for the DVD box in the net shops.

I have all of it on DVD, what a great series, incredibly original and outstanding performances and production. I just wish I could get my girlfriend into it - I've tried, and she crinkles her nose up and tells me to turn it off within 2 minutes. Just too bizarre for her. 

sharke

Kalle Rantaaho
I'm very, very surprised by the absence of "The League of gentlemen" in the lists suggested above.
That was one that kicked off the sofa. Actually, I'm just looking for the DVD box in the net shops.

I have all of it on DVD, what a great series, incredibly original and outstanding performances and production. I just wish I could get my girlfriend into it - I've tried, and she crinkles her nose up and tells me to turn it off within 2 minutes. Just too bizarre for her. 

When I get the DVD, that'll be an ordeal to me as well, :o) I'm almost sure it's too tilted for my better half.
She was out of the country when the show was broadcasted here, and obviously it was too tilted for the prime time audience here as well, because I can't recall it ever been repeated, whereas all the other britcoms are running all the time on many channels.

Yeah it was deliciously disturbing. Especially that German teacher with a fondness for young boys. My favorite was the whole Pauline, Ross and Mickey saga. Pauline's pens! "Everything I know about people, I learned from pens. If they don't work, give 'em a shake! If they still don't work, chuck 'em away!"

Lots of British TV comedy first saw the light of day as BBC Radio4 productions. So, on the radio side of things, all published by the BBC, I suggest

Cabin Pressure (co-starring Benedict Cumberbatch just before he was famous)
Old Harry's Game (start both of those from the beginning or they don't make sense)
I'm Sorry I Haven't A Clue (the pre Jack Dee era with Humphrey Littleton is decades worth of classic idiocy)
Yes Minister and Yes Prime Minister (TV but also available as audiobooks).
And, modern TV in the same vein, The Thick Of It.

bitflipper
Nobody's mentioned one of my all-time favorites yet, The IT Crowd.
 
I recently finished re-watching the entire series a second time and was surprised how hard I laughed at the repeats, despite knowing all the gags in advance.

That's one of my favourites too.
 
I like the scene when one of the says he doesn't know how to behave at a funeral when greeting the relatives of the deceased.  He gets told "Just tell them you're sorry for their loss, then move on".  Of course when it comes to meeting the relatives after the event he then says "I'm sorry for your loss, now move on".
 
Like you say you know the gags are coming in advance but they're still funny.
 
'Shameless' was one of my true favourites too but I'm not sure how that would translate without being intimately familiar with the lower echelons of the British social strata like what I am... ;)

jamesg1213
Loads, but a few of my favourites...
 
Detectorists (for me, the best British comedy in the last couple of years)
W1A (brilliant spoof fly-on-the-wall documentary about the BBC)
Twenty-Twelve (predecessor of the above, about the London Olympics)
Extras
The Office
Peep Show
Blackadder
Men Behaving badly
Auf Weidersehen, Pet
The Inbetweeners
Black Books
Cradle to Grave (new series, really enjoying it so far)
 
 
 

being a bit more metal, I will add:
The League of Gentleman (absolutely brilliant 4 man team over 3 series, macabre, bizarre comedy)
Green Wing (spoof hospital drama, some of the best UK comedy actors)
Spaced! (Simon Pegg's pre-Hollywood sit-com, if you like the Ice Cream trilogy you'll love this)
Psychoville (post-League of Gentleman. two series with Dawn French)
 

I've found that Americans are a lot fonder of some of the "sillier" British sitcoms than the British. Stuff like Are You Being Served, Keeping Up Appearances, that sort of thing. Back in the UK I used to HATE Keeping Up Appearances, I thought it represented everything that was wrong with the smug, upper middle class, luvvie darling BBC. But after living in the US a few years I came to love it and realized that Patricia Routledge is an absolute gem (along with the brilliant Geoffrey Hughes). 

Looks like I need to find a bundle of money so I can stay home and catch up on all of these!  I doubt that I've seen any of the ones mentioned above...