How to get out a stuck faucet cartridge without breaking it?

What people tried

Every workaround mentioned in the threads below. We haven’t tested any of them — and nobody here is claiming they worked.

1. 1
   Threading the handle screw into the cartridge and pulling with pliers
2. 2
   Putting the handle back on with the screw and pulling on the handle
3. 3
   Spraying penetrating oil and letting it sit for a couple of hours
4. 4
   Using an Allen wrench from the top
5. 5
   Using a Sawzall
6. 6
   Hitting it
7. 7
   Using a small pipe wrench
8. 8
   Using 2 pairs of channel locks
9. 9
   Replacing the shower arm as well
10. 10
    prying it out with a small screwdriver
11. 11
    threading thin wire or a zip tie around the back half to pull it out
12. 12
    carefully breaking the stuck cartridge apart in pieces
13. 13
    grabbing the brass stem with channel locks or pliers and pulling forcefully
14. 14
    Pulling straight out instead of twisting

In their words

Unedited, grouped by where they were said, most upvoted first within each place, each linked to the thread it came from.