Where are applescripts stored?

Using an Run AppleScript Action in Automator does not save an AppleScript file. Depending on what kind of Automator document type that you initially selected will determine its save location:

1. Services, or Quick Actions (prompt for a Service name, but not a save location as it is fixed)
   1. /Users/yourname/Library/Services folder with .workflow extension
2. Folder Actions (prompt for a folder action name, but not a save location as fixed)
   1. /Users/yournamne/Library/Workflows/Applications/Folder Actions
3. Workflows (prompt for filename and save location)
4. Application (prompts for name and save location, usually Desktop)
5. Print Plugin (prompts for name of plugin, default save location)
   1. /Users/yourname/Library/PDF Services
6. Image Capture Plug-in (prompts for name, default save location)
   1. /Users/yourname/Library/Workflows/Applications/Image Capture
7. Calendar Alarm (prompts for name, but default save location)
   1. /Users/yourname/Library/Workflows/Applications/Calendar
8. Dictation Command (prompts for name of Dictation command, but default save location)
   1. /Users/yourname/Library/Speech/Speakable Items

If you were never prompted with a Save dialog, then your Automator document was of types 1, 2, 5, 6, 7, or 8.

I guess the search function in macOS did not bring up any results either?

If you create a new script, will it it show you the latest folder you have been using, in case you use the editor to open a file?

Or show the "recently used" in the drop down menu?

Did you already check this guide? When you created that script, did you get the option to choose the folder, where the scripts should be saved to?

Mac Automation Scripting Guide, Saving a script