Internet router is Netgear cable as supplied by Telstra here. It has integrated WiFi and network switch.
There are 2 Ethernet connections from the device.
I certainly get you when you say it is difficult to solve a problem that you can't see happen. I have certainly seen the problem after it has occurred but still not been able to see it transition from a working to non-working state. I have identified that it is the print spooler app that stops, but not what is causing it to stop. After it stops, restarting the service temporarily fixes the issue. I spent 3 hours watching them one day last week and the problem did not occur while I was there. I have asked them to log what they do.
Are there any built-in logs that might help? I have turned on the operational print service log so will see if that tells me anything useful.
>>>Thanks for that information. It helps a lot.
Did you ask your Users to write down the error message with pen and paper? Or leave the screen on for you? Or take a picture with a Smarphone or Digital camera. At least one of your Users should be able to provide that level of cooperation. The log is a really good idea too, I've used that to help me track down many an intermittent problem such as yours over the years. As far as built in logs, no other printer logs that you've mentioned, but the print queue applets all tie back in to svchost.exe and the spawned services show up in Windows Event Viewer and System Viewer failures as number failure events, which you can then look up on the MSDN Knowledge Base online. Generally, it takes some advanced computer skills to interpret these, so I'm going to assume that you are aware of what these are and how to use them. Additionally, another good tool you can download for free is the PROCMON app (you can google it), and post back the resulting logs, neemo is quite skilled at interpreting the results from this app and recommends it frequently. This tool is a fine-tuning microscopic inspection version of the usual Windows event viewers I mentioned, and allows processes including the snaky ones spawned by svchost.exe to see where they are going what other processes are being generated, or failed, or called such as a missing or outdated .dll library. These often get hit by viruses/malware; so it's a good idea to scan every PC on your LAN. SAFE MODE scans
are best as you are aware, use whatever built-in AV scanners are already installed on those PCs, such as Norton, Defender, McAfee, Avast, etc. and ensure they are clean. You'll also need to check for RootKit and BootKit viruses using TrendMicro's RootKit Buster; free online at trendmicro.com. Also, you should download the free MALWAREBYTES from Malwarebytes.org if you haven't already installed and are using that to check for spyware viruses. Many spyware viruses and E-mail based jscript viruses target your print queue applets and either corrupt them or disconnect them from providing proper printer operation.
<<< Interestingly when the system is printing normally, the Admin printer log tells me that any job printed to the local HP 1020 printer has failed. Yet it tells me that the correct number of pages printed and the jobs physically do print OK. Is this likely to be connected to the issue in any way?
>>>I would agree with that hypothesis. This is often due to virus/malware infection, Windows registry/hive corruption, or hardware failure; usually read sector errors on aging hard drives or bit errors in failed RAM memory sticks. You should conduct full hardware testing on each and every PC on your LAN that accesses any or all of the printers you have connected. Here's a link to do the hardware testing I wrote step-by-step as well as subsequent Windows troubleshooting and all repairs up to and including drive wiping and full Windows reinstallation from factory Recovery discs: Windows 10 - Unclickable Task Bar.
If this were my LAN, I would do all of these things, as well as create a network diagram (hopefully I beat that one to death!), and then backup all Library data on each and every PC, and do a complete scratch Windows install on all PCs, and then retest for the problem. That's how I work. When I worked for IBM, Wang, Ford, Unocal, Lockheed, and other large corporate IT shops this is how things are done. It's also how I trained students who were becoming Network Engineers and Technicians to work too.
<<<
Are there any logs that I can turn on or view that track exactly what applications are being loaded and when? Maybe if I can marry up a particular application to the failure that will be a start.
>>>Yup, already mentioned, PROCMON, is a good one to start with. We have some talented people here who can interpret the results and guide you to possible solutions such as neemo.<<< The printers in the environment are:
HP OfficeJet Pro 86004 which is wireless-connected
HP LaserJet 1020 of which there are two; both locally connected via USB to different computers
HP 1102w which is wireless-connected
They also use print to PDF a lot using an application called Paperport.
>>>Very standard devices. Remember to disconnect all wireless printers from the LAN while troubleshooting. Print queue problems often originate from wired printers, not wireless, and without a $30k Lanalyzer at your disposal you're not going to be able to troubleshoot BOTH wired and wireless printers at the same time on the same LAN on the same subnet (which is how you've got your PCs and printers all connected)!!!
<<<
They use Office 365
>>>Again, very standard software; I used Office365 with Office2016 version here on my home LAN with none of the printing problems you mention and I have 1 wired color laser printer (or did) and 1 wireless Canon inkjet printer. All running on the same subnet through Ethernet switches and hubs with no problems. And all my PCs are now upgrade to W10 as well.
<<< All drivers are up-to-date and Windows updates are automatic on the computers.
For what it is worth the IP address scheme is 192.168.0.0/24 with the gateway as .1 The Netgear provides DHCP and there is an external DNS Server
>>>Also, very standard. If I was being sent in to fix your printing problem, I'd be charging about $600-$800 to produce the network diagram for this small LAN as the very first order of business. In Visio or Word or PowerPoint, I'd also put in all those devices along with their IP addresses and other nomenclature such as Make/Model of device, IP address, subnet information (which you don't have), and the topology layout. If this is a business customer, you should consider doing this and letting them know if they want their problem fixed you need to get paid for doing this part of the job, since it was obviously not in place when you got hired to work on it or fix it. 
That means that the business that hired you is not informed about such things, and if they didn't want to pay me for that part of the job, I'd walk away.
Just because the network is simple and connected together with just a few devices doesn't mean you don't need to apply Best Practices Troubleshooting methodology IMO.<<< Thanks again for any input
LP
Not sure also what kinds or sizes of networks you've designed and implemented, but if you've done any work for large companies such as Fortune1000 they have certain Standards that comply to Best Practices that require some level of network documentation. Especially if you are building networks for the Government or Military or Aerospace contractors as I have. If you never learned this in school or University, their curriculum is woefully under-preparing you.
Just my opinion of course.
This is a repeatability problem. If you can't make it occur, you can't fix it.
I would ask them to use a Smartphone or digital camera and take a picture of the problem, if you can't reproduce it. 
