For really doing a lot with the medium: Dresden Codak
I also like these, that I don’t think have been named:
Cat and Girl - a bit cerebral, I guess
Spinnerette - superheroics. In Columbus.
Misfile - OK, really a novel, read from beginning.
Flaky Pastry - just wacky, but pretty art.
Oh, & I should mention T. S. Wood’s Yet Untitled.
And Head Trip
Both of those are mostly gag-oriented.
What makes me laugh is the plot is so clearly not WoW any more, but it gets listed on WoW Insider’s Sunday comics roundup regularly.
Just remembered a webcomic I read that’s actually Everquest based (instead of Everquest/MMORPG spoof like The Noob), WTF Comics. Horrible, horrible at keeping up with the update schedule, though, but if you’re coming in new, there’s a fairly large archive to catch up with.
A few different titles can be found oncomicmix.com.
I just read some of Axecop and it is indeed awesome.
Of ones that haven’t been mentioned yet, I think the only one of my regulars is Piled Higher and Deeper, about life or the lack thereof in grad school.
I’ll add Being Five to the growing list. It’s a comic blog written from the POV of a 5 year old.
And Exiern ([noparse]http://www.exiern.com/[/noparse]) is getting better now the strip itself is doing well enough to not be thinly-disguised NSFW. I’ve no-parsed the URL because other bits on the page are NSFW
Girl Genius is a must!
Yet Another Fantasy Gamer Comic
Least I Could Do (slightly NSFW)
And, a promising newcomer well worth watching—
The Deadlys
Enjoy!
Well, to go through the better ones in my RSS feeds:
Amazing Super Powers: continuity-free, generally rather dark
Bad Machinery: essentially the sequel to Scary Go Round (itself one of my favorites)
Dead Winter: a zombie-apocalypse comic, pretty slow moving (without a lot of zombies, frankly); has some nice, detailed art if you don’t mind the style
GastroPhobia: comedic, well-drawn, about an ex-Amazonian warrior and her son in a farcical ancient Greece
The Meek: very, very pretty, tightly plotted, well-scripted
Necessary Monsters: a heist/horror movie mashup; very clever, takes itself just seriously enough
Not updated since last year, apparently, and not an ongoing story but a “comic strip”, I guess - but very clever and makes a very entertaining half hour to go through - A Simple Apology:
And a few more that I missed. It might be obvious to those who are familiar with the comics I mention, but I’m really quite a stickler for the art. Less-than-professional is okay, as long as the artist tries and does good work conditional on the skills available (see e.g. early/mid Scary Go Round; John Allison is a model of self-improvement, and Bad Machinery is as good as anything out there). I just don’t see anything appealing about – say – Order of the Stick.
Nobody Scores: Clever, surreal. Used to be one of my favorites but its author seems to have more or less given it up, possibly with some comics-are-crap related angst.
The Abominable Charles Christopher: Very cute, realistically-drawn-talking-animal related. Professional. The one-offs can be very good; the plot-heavy threads give the impression of the author not having any plan at all.
The Phoenix Requiem: It’s not my favorite comic by a long shot, but clearly a lot of care goes into its creation. The characters may be a little … plasticine, but I’m a sucker for detailed, careful environments, and she delivers.
Of the ones already mentioned by others, Dresden Codak, FreakAngels, and Gunnerkrigg Court all get my vote.
Multiplex - a comic about the staff of a multiplex theatre
This is because, unfortunately, he doesn’t. I can’t be bothered to find the cite, but he mentioned it in an interview early on in the comic. Even so, I really liked this comic for a while until the site was inexplicably blocked at my office and I lost track of it.
The two I read on a daily basis are the already mentioned Questionable Content(I love Screaming Bird BTW) and the yet to be mentioned Cyanide and Happiness.
Read the 1st 4 of this one and love it!
Looks like the strip is heading towards an interesting sceneThere are three parallel plotlines of women becoming interested in Gary. So I’m anticipating a big scene where Amber, Sonya, and Yuki all put a move on him at the same time.
Some good stuff already mentioned. I’ll add the completed webcomic A Miracle Of Science. It starts out B & W but goes to color at Chapter 2. It’s set in a world beset by Mad Scientists, with a cop who specializes in hunting them down paired with a Martian partner for a particular case. “Martian” in this setting referring to a rather cute young woman from the Martian group mind, which is more technologically advanced than everyone else. Unusual in fiction for being a benevolent group mind.
Actually it didn’t even miss an update then; an “Emergency Schlock Mercenary Backup Page” was set up for the duration without missing a beat.
That’s the comic my username comes from btw.
There was a beat missed, just maybe not a day’s worth of beats. 
This has been really educational so far! I’m especially looking forward to the story-arc based comics, but all the strips are going to be very interesting as well. Thanks a bunch to everyone!
A good site for finding webcomics is The Belfry. You can search by genre, update frequency and it has a PG-13 filter, if you want to keep it clean.