Find a list of installed apps that can no longer run because they have not been updated with the current operating system?
iOS

Support for 32-bit only apps was completely dropped in iOS 11. If you upgrade from iOS 10 to iOS 11 (or iOS 12) and have apps installed on your iOS device that are 32-bit only, the list for the same can be obtained by going to Settings → About → Applications.

If you are running iOS 10.x, iOS 10.1 started presenting alert to users of 32-bit only apps that the app needs to be updated to be usable in future release of iOS.



Source: iOS 10.1 Reinstates 32-Bit App Alert, Warns Older Apps May Slow iPhone



macOS

Starting with macOS High Sierra 10.13.14, Apple started alerting that 32-bit only apps will stop working in the future updates. Upon launching such an app for the first time (after the macOS update), user is alerted with this dialogue:



The list of all 32-bit only apps on macOS can be obtained by going to Apple Menu → About This Mac → System Report... and in the System Information window that opens, go to Software → Applications in the sidebar. In the list of apps presented, click on the 64-Bit (Intel) column to sort and see that entries with No.



It was widely expected that macOS High Sierra may be the last macOS release to support 32-bit only apps and just like what happened in iOS, such apps will stop working in macOS Mojave 10.14. However, Apple later chose to indefinitely postponed the decision and 32-bit, as of now, continue to work in macOS Mojave 10.14.

Apple support document, 32-bit app compatibility with macOS High Sierra 10.13.4 is work referring. Quoting from the document:


  When will the 64-bit transition be complete?
  
  The 64-bit transition for macOS and macOS apps is still underway, so final transition dates have not yet been established. But now is a good time to check with the software developer to see if 64-bit versions of your favorite titles are available.





  How do I check if an app is 32-bit or 64-bit?
  
  From the Apple menu, choose About This Mac, then click the System Report button. From the system report, scroll down to Software, then select Applications. When you select an individual application, you will see a field titled ”64-bit (Intel)”. “Yes" indicates 64-bit; “No" indicates 32-bit.




Apps for macOS can be distributed via Mac App Store or independently by the developers. Starting January 2018, Mac App Store stopped accepting new apps and starting June 2018, stopped accepting updates to the existing apps which lack support for 64-bit.


  As a reminder, new apps submitted to the Mac App Store must support 64-bit starting January 2018, and Mac app updates and existing apps must support 64-bit starting June 2018. If you distribute your apps outside the Mac App Store, we highly recommend distributing 64-bit binaries to make sure your users can continue to run your apps on future versions of macOS. The last macOS release to support 32-bit apps without compromise is macOS High Sierra.


Although the last line of the developer news quoted that 32-bit app support will be dropped without compromise post macOS High Sierra, that decision appears to have been retracted indefinitely for the time being.