Why does the search function on my desktop no longer work ?

How to rebuild the Spotlight index on your Mac

If searching your Mac doesn’t return expected results, rebuilding the Spotlight index might help.

1. Choose Apple menu () > System Preferences, then click Spotlight.
2. Click the Privacy tab.
3. Drag the folder or disk that you want to index again to the list of locations that Spotlight is prevented from searching. Or click the Add (+) button and select the folder or disk to add.
4. To add an item to the Privacy tab, you must have ownership permissions for that item. To learn about permissions, choose Help from the Finder menu bar, then search for “permissions.”
5. From the same list of locations, select the folder or disk that you just added. Then click the Remove (–) button to remove it from the list.
6. Quit System Preferences. Spotlight will reindex the contents of the folder or disk.

Sorry, need to use sudo:

sudo mdutil -E /

You must be using an Admin-enabled account.

It will ask for a password. Just enter the password for your account. It won't show anything when you type it. You just have to type the password and hit return.

Try this, and see if the Spotlight rebuild (once the rebuild completes) works better: How to rebuild the Spotlight index on your Mac - Apple Support

Long shot, in Sys Prefs>Spotlight, uncheck Bookmarks & History, Rebuild Spotlight Index again.

Try entering this command in Terminal. Just copy and paste.

mdutil -E /

It should do the same thing as what MrHoffman suggested, but some have reported it works whereas the GUI method does not.

Open Activity Monitor and Sort by CPU usage. Does mdworker, mds, mds_stores show up using a lot of CPU?

If so, it may be hung up trying to index something. Usually erasing the index will allow it to get past the blockage.

I've given up trying to search with Spotlight, I find the free EasyFind a much better alternative.

https://www.devontechnologies.com/products/freeware.html

Find Any File...

http://apps.tempel.org/FindAnyFile/index.php

Hold Option or alt key when selecting Find to Find All.

Yeah; I used to think that the nuke-and-pave path was a drastic approach too, then I ended up having to go that route after some mishaps with a beta that shall remain nameless, and learned how straightforward it all was—once you have Time Machine or other backups, and younreally want those running at all times anyway—and how well the migration path worked, and how much got cleaned up by the migration.

Apple likes selling new Macs, and that means existing folks will need to migrate. And that’s what this path is.