> I know multiple people who reported their expiring passport as stolen, just to be able to keep it

Eh, you want to watch out doing that - some countries get real suspect if you have ever reported a passport stolen in the past.

Hmm, interesting. In my experience they just cut off a corner and put a large red EXPIRED stamp on the info page.

It varies per country. In the Netherlands you can request to keep the old document; it will be returned to you with a couple of ~12mm holes punched through it.

My girlfriend needed her old passport for travelling with an Interrail train ticket because the passport expired between the time of ordering the tickets (and submitting the document number) and travelling.

Well, your experience would be limited to the one or two countries you're a citizen of :)

It's definitely a thing across various ex-Soviet countries, and likely others as well.