Specifically, the transition problem includes the following important services which rely upon PSTN:
Fire and other alarm systems
And presumably other things, but these are kind of serious business.
Up until last year, I was still having to fax things every few months, usually involving financial services who couldn't seem to accept scanned and emailed documents. Once they could accept scanned and emailed documents, I started either scanning and emailing or, more recently, just taking a picture with the phone and mailing it from there. I tried to establish end-to-end digital workflow using HelloSign but it kept crashing on me, so I still print the damn things out, sign them, take a picture and email it, which is embarrassing, but whatever.
Health care still relies extensively on fax (particularly in the communicate-to-pharmacy stage, bizarrely), but that's changing with regulations involving EMR.
There's an odd thing involving letters of intent and sports teams using faxes, but that's changing and besides, it's a niche weirdness along the lines of wedding telegrams.
And there are still a ton of small professional practices (often law, finance/accounting and similar) which rely heavily upon fax communication. They are the only group that really relies on faxes that isn't experiencing any organized pressure to convert and thus they are the one likely to be most heavily impacted by the end of PSTN. Good news: lots of these people are voluntarily switching away from traditional land lines/copper/PSTN, in favor of VoIP for cost savings. When they switch, they learn that their fax doesn't work so good over their new phone system and frequently decide to go to an online fax service, altho some presumably sign up for FoIP. FoIP directly packetizes fax signals and ships them over the internet, rather than sending them as analogue signals that get (incorrectly) packetized as part of the VoIP packetization, so FoIP works well -- and doesn't rely upon PSTN.
It looks like most fire and other alarm systems work roughly the way that fax machines do: they _were_ sending out signals in analog form and the result interacts poorly with VoIP, which optimizes its packetization to produce a good sounding human voice. The solution will be the same as with fax machines: replacement equipment that packetizes the signals and sends them out over the internet without involving the PSTN. In the meantime, most of these things can work over cellular systems.
I'm still poking around at how 911 will handle the transition. Right at the moment, it's looking like not great, but I feel confident that technical problems have technical solutions.
(ETA: A little bit about 911 and VoIP. Power outage issues apply to users of Digital Phone/Digital Voice services as well. http://www.fcc.gov/guides/voip-and-911-service)
Will this be free? Nope. Will the end result be an improvement? Probably. Will the 20% of the population which is pugnacious in its opposition to all change see Evil Conspiracy in all this?
ETA: A possible future: http://www.nojitter.com/post/240145831/finally-texting-to-911
Another possible future: http://www.hightechforum.org/a-broader-vision-for-wireless-9-1-1/