Everyone knows the story, even if, unlike me, they are not old enough to remember it: the video format VHS overcame the superior Betamax format to dominate the 
home video market in the 80s and 90s.
Of course, the claims that Beta was superior are rather tendentious but the fact that video producers stuck with a Beta format for their own tapes long after the rest of us switched to a VHS monoculture must say something.
Now, inside your browser, the same thing has happened. As is noted in today’s Guardian, the 1987-vintage GIF format refuses to die. These days you see far fewer static GIFs than even a few years ago (though they are still out there in large numbers) – JPEG and PNG dominate. But you’ll have to look very hard for an MNG (the ‘official’ PNG analogue of the animated GIF) or even APNGs, the unofficial but more widely used attempt to animate PNGs.
A few years ago I wrote a Perl PNG module – Image::Pngslimmer – which replicated many of the functions of the C libpng library, so I could use some of that in CGI code without having to switch from Perl to C and back again. Then – this was 2006 or so – PNG support was quite weak in browsers and GIFs were far more plentiful.
PNG is a superior format to GIF (especially for line drawings and similar – for general photographs JPEG is the superior choice, unless you truly demand a lossless format) and it is a good thing that it has edged GIF out in many places. But it seems we will be stuck with GIFs for many years yet.
Related articles
- Google and GIFs: You Can Upload But Ye Shall Not Animate (cogdogblog.com)
- How to Save Images as JPG or PNG Using the Gimp (simplehelp.net)
- PNGGauntlet Compresses PNG Images Without Sacrificing Quality [Windows] (makeuseof.com)
- VHS and Why It’s Awesome – A UK History (grizzlybomb.com)
- Are We Approaching Peak GIF? (theatlanticwire.com)
cartesian product 
I wonder how many “legacy” web pages are out there with GIFs in them. Some of those pages are still valuable, even if not maintained; so we’ll still need rendering support for GIFs for the indefinite future, even if at some point people stop creating new GIFs. I see similar problems with other web technology upgrades (HTML5, the never-quite-materializing semantic web, etc.).
I’d guess there are probably more GIFs out there than PNGs even now. Though JPEG has plainly taken over for most things (even for line drawings, where it is lossy overkill compared to an optimised and lossless PNG).