This site has the best granularity for recent PlanetMath outages that I found so far.
There is a GitHub developer issues page for bug reports on the PlanetMath software, but it has been moribund since 2014, possibly because of software revisions in 2015. The Google Groups Planetary Developer forum is similarly moribund.
I reached out to Aaron Krowne, one of the PlanetMath founders, by email to see if more information is available. He replied promptly and CC'd a number of individuals most recently involved with the site's design and maintenance. Joe Corneli responded with information that included the Dec. 2016 site backup on GitHub.
Aaron mentioned that discussions have been underway for several months to "[get] the content up in a new environment." I'll try to stay involved and report back here when we could decide about efficiently revising dead links within existing Math.SE posts.
Update(01/31/18): We should have an estimate "in a few days" of WHEN "the dead links should come back to life." (Turns out they are only MOSTLY DEAD.) Discussion of whether extensive edits will be needed or desired can wait for that.
Update(07/09/18): The PlanetMath site came back online in March, 2018, and I optimistically thought that most of the links here would return to working. However (prompted by a recent Comment from Alex W.) sampling of these links shows that the simplified structure of the site probably leaves a large number of the existing links broken. A query shows about 800 links to
planetmath.org. I've seen a couple of patterns in older versions of those links which won't work: linking to a folder
planetmath.org/encyclopedia (189 cases) and linking to a folder
planetmath.org/sites/default/files (25 cases). I'll refine my queries to give more detailed counts.