Storage of luggage on 1st vs. 2nd class trains

Depends on the train. If you have airline carry-on sized bags, in either class they can fit over your seat in the rack. On some trains, sets of seats back-to-back form an A-frame and you can tuck your (not too big) suitcases in there. Most cars have a luggage rack at one or both ends of the car where larger bags fit.

First class carriages have more space overall but not necessarily at your seat. If you are worried about theft, get up when the train is stopping and make sure you can see your bags or be close enough to protect them.

It depends on the train and the size of your suitcase.
My carry on bag with wheels 9X14x22 inches fit in the overhead space or under the seat on the Belgian Swiss and Italian trains. I've had to leave it in space by the door on French TGV trains

Differences between 1st and 2nd class are not usually in the design of the luggage space, but in the number of people sharing it, and the number of seats across each row (3 vs 4). Storage designs vary more by the age, country, and speed/brand of the train, and sometimes whether it's aimed at commuters or long-distance travelers.

We put our luggage in the racks above our seats.

On our trip a couple of weeks ago from Brig to Venice (second class) our bags did not fit in the overhead space. We have small-ish carryons (RS rolling backpacks, eBags Weekender backpacks). The bags were too thick. There might have been 6/7” or so of room on the overhead rack, enough for a briefcase. We stuffed our bags under our seats; they kinda fit and we used them as footrests. All the hard sided spinners were in the aisles because the racks at the end of the car were full. It was a mess.

Be prepared to be creative — you never know what you’ll get.

There is no difference in the luggage storage arrangements between 1st and 2nd class.
There are three possible place you can store your bags on a train:

1) Overhead racks. These are larger than plane racks, and are just shelves, no doors (trains rarely experience air turbulence ☺)
Click here for photo

2) Between seat backs. If two seats are back-to-back you have an A-shaped space between them. This is the best place for big or heavy bags.

3) On some long distance trains only extra floor to ceiling luggage racks either part way down or at the end of the carriage. This is the only place that might be out of sight.

For more info and photos see here: https://www.seat61.com/luggage-on-european-trains.htm

"And I've also read/ heard warnings about luggage being stolen on trains, even if you secure them with a chain lock." - This, like many other things, is greatly exaggerated. If you use a lock (and the locals don't) be careful not to block someone else's luggage being removed before yours, or prevent bags being re-arranged to fit better..

"But in 1st class you can leave your suitcases next to where you sit." - 1st class makes no difference. Do not put your bags in the aisle or on a seat, that will annoy people.

We had no problem finding places to put our luggage in second class, usually overhead. We had standard size carry-ons and rarely had them out of our sight. The exception was on some smaller trains, like from Grutschalp to Murren, where are you were supposed to leave your luggage at the end of the train, but the train is so small and everyone is getting on and off at the same stop so there really is no way people can steal it.

Thank you to all of you. It has been very helpful. I think this travel forum is great.