Newspaper comics in general since the 1970s are Deader than Disco, though not as fast as other (past and present) newspaper sections once popular. Few are running as new strips drawn by the original artist; fewer were started after 1995 or so, mostly because of Values Dissonance, of course. (To quote South Park: "Only gay little dweebs read the funnies, Butters.") There are two other major reasons for this decline, and both of them have to do with the Internet. The first is the rise of webcomics, which have the Infinite Canvas instead of the papers' ever-shrinking panels, a simpler method of publication (updating your own Web site vs. signing deals with newspapers), and less censorship. The second is the decline of newspapers themselves in the face of New Media, which means fewer people reading the funny pages and thus less money to be made in writing for them than there was as late as the 1940s, when successful comic artists like Chic Young, Al Capp, and Milt Caniff could make upwards of $100,000 a year. Most major newspaper comic artists make middle-class incomes through a syndicate now (the rewards of signing a contract far smaller than in other industries); for a new artist still working a day job, getting there from here through indie channels is less daunting.
The newspaper comics generally considered to be the greatest of all time are The Far Side, Peanuts, Calvin and Hobbes, and Bloom County. The fact that none of these are currently running (Peanuts does persist in reruns) is definitely a factor (their creators retired under the condition that no other cartoonist replaces them). There are only a handful that have any sort of esteem online, most notably Dilbert, Get Fuzzy, Pearls Before Swine, and FoxTrot (and even those four have vocal hatedoms, especially Dilbert). Others that are somewhat less popular, but still popular online, include Zits, Retail, and Liō.
"Legacy" strips have especially suffered for this, especially because of their refusal (from either authors or syndicates) to keep up with modern times, especially after the success of Doonesbury and other comics during the 1970s. The fact that The Katzenjammer Kids, the longest running narrative of all time (1896–present) doesn't yet have an article on this Web site just shows how far they have fallen.
In 2009, DC Comics did an experiment called "Wednesday Comics", in which they published a weekly "newspaper", consisting of nothing but comics done in the format of the 1930s; I.e., they were full color, 14″×20″ (35cm×50cm), some of them fully painted. They were written and drawn by some of the biggest names in the industry (Gaiman, Kubert, Simonson), and starred both DC's greatest hits (Superman, Batman, Wonder Woman) and quirky fan favorites (Kamandi, Adam Strange, Metamorpho). The experiment lasted three months. Alas, newspaper-style comics don't sell.
Those who didn't grow up in the 1980s might be amazed at how ridiculously popular Garfield was — nearly any product you could think of had the cat's face on it. The strip was even considered edgy and controversial at times, which may come as a surprise among those who remember it only for endless jokes about Garfield being fat and lazy, Jon being a total loser who can't get a date (until he and Liz became an Official Couple), and Odie being dumb as a sack of hammers. The strip's Merchandise-Driven nature died down somewhat in the 1990s, but hasn't stopped fully. Even so, Garfield is now known mainly for three things: 1.) the aforementioned fat/lazy/loser jokes that many consider long since done to death; 2.) Lyman (Odie's former owner) being the poster child for Chuck Cunningham Syndrome on the comics page; and 3.) overall being the strip that people love to hate. It certainly doesn't help that the strip's art style hasn't changed at all in about a quarter century, and the Story Arcs petered out sometime in the mid-1990s, with Garfield and company now just phoning in "one-a-day" gags. The strip's decline to Snark Bait territory has spawned humorous remixes such as Garfield Minus Garfield and Square Root of Minus Garfield. The fact that Garfield creator Jim Davis was not only approving of the former, but even submitted a few of his own to the book, really says something.
Serialized dramatic or adventure oriented comics. Popularized by Hal Foster's Tarzan, Alex Raymond's Flash Gordon, and Soap Opera comics such as Nicholas Dallis' Rex Morgan M.D. and Apartment 3G. If you're of a certain age, you'll also remember that once there was a Star Wars daily newspaper strip that lasted several years. The four (maximum) frame a day strip format has proven more suited to self contained gags than it is to a continuing story. This is one of the reasons that the Comic Book was invented, to decompress the art layout, give more space to slower paced, dramatic stories that the strip format doesn't afford them. Also, space for comic strips has been shriking in newspapers resulting in smaller reproductions of the strips themselves. This isn't much of a problem for gag strips which are generally rendered in a simplistic style. However, most adventure and drama oriented strips are drawn in a realistic style that doesn't hold well to shrinkage. Other than legacy titles like some of the aforementioned (many of which now have more limited syndication than they used to), the serialized adventure/drama comic strip seems to be a relic of the past.