I have documents I would like to have stacked on my desktop.
Specifically, to group all files with a particular extension all together using Mojave's new Stacks feature.
Is there a way to register a new stacks kind?
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No - Apple designed the default configuration but does allow you to stack on tags, which are user definable.
You could also make a smart folder to locate all the specific files you want, and then curate the tags easily, but not automatically out of the box.
Here's how to use tags for custom desktop Finder stacks.
In the image below, you can see the option under
Group Stacks by -> Tags.
Each of the files you want to put in this Stack must have the same Tag. When you do this, macOS will create a custom Stack for you for all files that contain that Tag.
This is the only way that I'm aware to create a custom Stack.