Passport Stamps

Nope. You only get your passport stamp upon entry and exit when going through immigration for the Schengen zone. There are no border checks with the Schengen zone. Like going from Indiana to Kentucky. If it is a country outside the zone then there will be an exit and entry stamp because you have to go through immigration. Your passport is not a souvenir for collecting World Stamps at Disneyworld. And you absolutely do not want your passport littered with extraneous stamps.

No, not possible when travelling between Schengen Area countries. Your passport will be stamped once when you enter the Schengen Area (assuming you are not a EU/Schengen Area citizen); and once on leaving. Travelling between Schengen Area countries there are no passport checks, same whether you fly, cross the border by train, bus, car or on foot. Trains do not stop at the border.
Between other countries there will be passport checks and stamps.

A passport is a legal document, like a birth certificate. Where and when it is stamped is governed by international treaties and national laws.

Travelling between Schengen Area countries there are no passport checks, same whether you fly, cross the border by train, bus, car or on foot.

That's technically not true, although the answer to the OP's question remains the same -- you can't get your passport stamped going from country to country w/in the Schengen zone. Our passports were checked last year and this year on arriving in Lyon from another Schengen country, and last year upon leaving, due to the state of emergency in France. But it didn't seem to be the same kind of passport check as when we entered Schengen; the passports were just looked at in a cursory fashion but not run through a machine or stamped.

PPs have answered the question insofar as the Schengen countries are concerned, but as one PP mentioned, your passport should be stamped on entry and exit for the non Schengen countries.

The main motivation of the OP is wrong. Schengen stamps are virtually identical country to country so one entering and one leaving is more than enough.

Our passports were stamped when we took the Eurostar (train) from London to Paris.

And you absolutely do not want your passport littered with extraneous stamps.

Why not. What’s wrong with having a passport full of stamps.

Why not. What’s wrong with having a passport full of stamps.

Nothing, except that if you run out of pages before the passport expires, you have to pay for a new passport. Extra pages are no longer issued. Also, some countries require blank pages upon entry. For example, entry to South Africa requires two side by side blank pages.

Our passports were stamped when we took the Eurostar (train) from London to Paris.

Yes, because you were entering the Schengen Treaty Zone. Bet you also got one when you left.

There is of course, the problem that nobody in any official capacity has a stamper available when you cross an international border within the Schengen Zone. They'd love to stamp you passport, but they didn't get issued one. But cheer up, I hear if you go to the Tourist Office in Liechtenstein, they will stamp your passport if you pay them 5 CHF.

the problem that nobody in any official capacity has a stamper available when you cross an international border within the Schengen Zone. They'd love to stamp you passport, but they didn't get issued one.

There is nobody at Schengen internal borders, with or without a stamp.

@Robert, yes France is currently doing ID checks at some airports. This is not a full passport check as you are not stamped in or out. Legally this is the same as a policeman asking for your ID in the street. This has been happening since the current state of emergency started. AFAIK it is due to end in a few months. And at airports you always need to show your passport as ID, to see it matches the name on the ticket.
Land borders (train, road, on foot) do not have any checks, nobody is there.

As Sam says, the UK is not in the Schengen Area, so the Eurostar train from London to Paris is entering the Schengen Area, and you do have passport checks and stamps.

And you absolutely do not want your passport littered with extraneous stamps.
Why not. What’s wrong with having a passport full of stamps.

A passport is an official document. On UK passports it says (from memory): "This passport remains the property of the government". Putting unofficial stamps in it is defacing a document and can result in the passport being refused at borders. And it might result in running out of space in the passport.

My expired passports are retained as souvenirs and it's fun to look back and recall countries visited and when.

This is not a full passport check as you are not stamped in or out.

Yes, as I mentioned, I'm pretty sure our passports were not run through a scanner or other device. In fact, I think when we flew out of Lyon last year (to Amsterdam), the immigration official didn't even open our passports. It makes sense, since we were leaving France. And this year, although we underwent a cursory passport check on arrival at Lyon from Munich, there was no check of any kind when we left.

If my trip includes going to the UK and then to the continent and the return to SFO or OAK, including backtracking, I want every passport stamp possible. I know that upon return at SFO or OAK the passport is not stamped anymore as it was , say 15 years ago. Back then my US passport was always stamped, not any more.

After the few perfunctory questions pertaining to the trip, eg, did you travel alone (yes), etc, and I'm told that I'm set to go, and if my passport is still not stamped, I ask for that. They are always willing to oblige. In the end I get what I want...another US passport stamp.

@Fred, most countries do not stamp passports of their own citizens (no point, they can stay as long as they like). And EU/Schengen countries do not stamp passports of other EU/Schengen countries (same reason). No questions are asked, you have a legal right to leave and enter your own country, questions are pointless. I haven't got a stamp from a European country for decades.
If passports were stamped, some people would need a new passport a lot quicker than 10 years.

All my US passports were stamped as a matter of course upon my return, which was always at SFO, ie, after the perfunctory questions, the Immigration person stamped the passport. Now, after the perfunctory questions, the s/he closes the passport and you know s/he is finished with you.

At that point I ask, "can you stamp it?" They have always been willing to oblige; never have I had an Immigration person say something to this effect, " you don't need it" or "why?" I get the "Admitted" stamp...always. That's what I want.

Your passport will be stamped in the following scenarios: 1. Entering the Schenegen area and leaving it. 2. Entering the UK or Ireland and that's it for Europe

It is important to get an entry and exit stamp in your passport especially if you make multiple trips to and from Europe. Helps to easily allow for counting how many days you have been in the Schengen zone so you can see if you are getting close to your maximum. Yes, this information is probably electronically stored in some computer somewhere in Europe, but is is easier to just look at the stamps and do the math when you need to know.

But as everyone else said, the days of getting a stamp in your passport for every European country you visit are gone unless you fly in and out, to and from, non Schengen countries.