How delete messages that appear in Phone Link but are not on my phone?

On your (Android) phone, even if you send something to the archive or spam folder they will be gone from your phone messages. On the PC in Phone Link they will still be in the list of messages and you can't delete them.

On the phone, in Google Messages (I don't know about Samsung or other default messages app) when you see the list of messages, at the top, next to the magnifying glass is a little icon of a person. Click on that and it will take you to a menu where you can select Archive and Spam messages. (Swiping left or right on a message will archive it.)

(I miss Pogo's wisdom)

What spam folder and where - PC or cell phone?

I have never been able to delete messages in Phone Link. I don't see any option to do that. I've always deleted them on the phone. If they are deleted on the phone but still exist in Phone Link then (ON THE PHONE) you have to go to the Archived or Spam & Blocked section and delete the message. You reach this area (at least on my Samsung S24+) by touching the "face" icon next to the Search magnifying glass, then select Archived or Spam & Blocked and delete any you find there. Then they will be gone in Phone Link.

The 3 dots (ellipses), shown in the Phone Link message listing will give you Pin, Mute & Hide, nothing else.

When I click a message in my PHONE LINK and then on the 3 dots (...), there is no Delete/Trash.

We should not have to deal with the same unwanted message in Phone Link more than once. I should be able to do a DELETE ALL somewhere, whether it is on my Phone Link app on my PC or link to windows on my phone. I am using Google Messages.