Spec: http://www.w3.org/TR/html-longdesc/
It's obsolete in HTML5: https://html.spec.whatwg.org/multipage/obsolete.html#attr-img-longdesc
According to http://reference.sitepoint.com/html/img/longdesc , which is admittedly old, only Firefox supported it, and then only partially.

@cvrebert : FireFox currently support it partially and is extending it’s support. It is not yet supported in the ᴀᴘɪ.
Presto based Opera fully support it. Internet Explorer support it since version 8, but it need to be enabled with a registry key.
Wanted to find the answer for blink based opera and webkit bowers based on http://caniuse.com but I couldn’t find it.

Google Chrome use an official extension https://chrome.google.com/webstore/category/collection/accessibility which might get native support in the future.

javascript: ᴜʀɪ aren’t supported by Firefox. data: ᴜʀɪ aren’t supported internet explorer.

The attribute isn’t obsolete it’s a recommendation.

What about Microsoft Edge?

@cvrebert : I wanted to know about it on caniuse.com.

@cvrebert : I found official informations for microsoft edge : https://msdn.microsoft.com/en-us/library/mt346148(v=vs.85).aspx http://download.microsoft.com/download/8/4/2/8427CF1B-08B3-4557-952D-102E7A8FA64C/[MS-LONGDESC].pdf http://download.microsoft.com/download/8/4/2/8427CF1B-08B3-4557-952D-102E7A8FA64C/IEStandDoc.zip

Sorry to contradict @cvrebert but it is back in 5 : http://www.w3.org/TR/html-longdesc/ With all the new accessibility laws in EU and the 508 in the States made no sense to make it obsolete. Developers just have to learn to use it properly, like I have to do! :-(

Wow, this argument of whether or not longdesc should be used has been going on for years, and it isn't a feature article on Can i use yet? Granted, the longdesc attribute was added, removed, then added back into the W3 specs again, but we should be giving it more thought, and letting people know which browsers support it, and how. Even if it shows red across the board, it would act as a wake up call to web browser authors.

+1 Tracking support of this would be great.
See also http://diagramcenter.org/standards-and-practices/html-standards.html
which states it is supported in at least Firefox, IE, and Chrome.