Please ONLY add to this thread other repositories; general information about AppImages, Snaps and Flatpaks, techniques for working with them and corrections of any mistakes in this thread,
To assist anyone looking for a particular application or type of application, please discuss specific applications under the Software Section discussing that specific type of application.
At its best, an AppImage is an application which is self-contained and fully portable. That is, it does not look to the operating system under which it is runs for any library or other dependency; and it can be run from anywhere: no need to install it.
The usefulness of such devices have been demonstrated by Frugal Puppy's employment of SFSes for a decade. And when they appeared even the first debiandog supported SQUASHFSes. Separate from the rest of the operating system, library conflicts resulting in failed or broken applications can be avoided: If an application has libraries which conflict with the libraries of another applications, just don't load them at the same time. When not loaded, SFSes/SQUASHFSes make no demands on RAM or CPUs.
Currently, at least in theory, Puppies and DebianDogs can make use of AppImages, Snaps and Flatpaks. These provide the above benefits, and more. When an SFS is loaded, its component files are distributed throughout your system; hence the possibility of library conflicts. AppImages, Snaps and Flatpaks only look within their own structure for libraries. This not only avoids conflicts between applications. Some libraries are required by the operating system, itself. Attempting to use a conflicting library even temporarily would result in operating system failure. Such conflicting libraries can be built into the AppImage, Snap or Flatpak.
Perhaps as important today with the thread of malware present whenever access to the internet is had, these methods of application deployment are sandboxed and not easily modifiable. Sandboxed means they do not interact with your system or other applications. To modify them, you have to unpack them, make changes and repack them. Any threat is at worst temporary: existing in RAM and wiped on shutdown.
Of the three, AppImages are the most useful for Puppies. To make use of Snaps and Flatpaks [except as a source to be unpacked and rebuilt] requires that the applications to access their repositories be installed. Many AppImages, on the other hand, can be used OOTB. In fact, hopefully fredx181's utility for creating a 'simplified' AppImage will soon be available on this Forum.
Unlike SFSes/SQUASHFSes, AppImages at not linked to the operating system thru /mnt. Rather, they link thru /tmp. The contents of /tmp are not preserved in SaveFiles/Folders on shutdown/reboot. This may pose present a challenge if you wanted to use an application in AppImage format as your default application for its type of work. It isn't difficult to provide a menu entry or launcher. A bash-script "on the path" could call it and can be used in Rox's right-click menu. But assigning mime-types to it so that Left-clicking a datafile would open the Application-AppImage is something 'above my paygrade'.
Not all AppImages will work OOTB. Some never will. They are built and tested under "Big-Boy distros". Puppies may not include some libraries which the builders assumed would be present. It is possible to unpack an AppImage, include missing libs and repack them. But, especially when the missing library are Qts or Qt5s, it is easier to install those libraries into Puppies so that they will be present not merely for a particular AppImage, but for any application which needs them.
Some AppImages refuse to run 'as Root'. Mike Walsh showed that those could be opened via a wrapper.
https://appimage.github.io/apps/ is a source for many AppImages.
Edit: per bigpup @viewtopic.php?p=14706#p14706
"Slightly different link to the one on the first post above.
https://appimage.github.io/.
AppImageHub main web page divided into sections and having visual images of each appimage."
Another entry-point providing categorized links to AppImages. https://www.appimagehub.com/. Thanks, bark-woof-fetch, for calling it to our attention.