@ChristianJ suggested this topic and I agree, it is a good one that had a little traction in the “Classic Forum” as well, so I am starting it up again here.
Personally, I keep all postcards that are handmade, include a sketch or something similar, have a personal message, have an image that catches my fancy, or where it’s clear someone really went through some effort to match my profile. That covers most cards I receive, but still I end up with cards I don’t want to store for all eternity!
When I receive a postcard that I decide not to keep, I follow this triage:
If it has an interesting stamp I give it to a colleague who collects stamps with his grandson.
If not, but I think I can recycle part of the image for a craft project I will save it for that.
If none of the above, it goes into the recycle bin.