Backpacks allowed - museums and attractions

In principle, no. Most museums do not like back packs of any size. People turn and knock things over.
Consider a small, cross body bag instead?
Most museums have lockers you can use. Put in a coin, get it back when you open it. 1-2€ coins depending on the country. Their websites should have this information available to you.

I second the vote for a small crossbody bag. I wore a Patagonia Atom sling bag in England and Scotland last June. In Italy, in October, I wore a Tom Binh small café bag. The only time I had to check either bag was at the Borghese Gallery in Rome.

I think it depends on where you are. I was recently in England for over 5 weeks, and visited dozens of museums and castles and stately homes. I wore a small backpack (7L) and the only time I had to remove it was at Cragside House in Northumberland. And even then, I had the option of just carrying it (which I chose) or wearing it in front.

I use a backpack purse, to hold all my much needed stuff: maps bottle of H2o etc. I take it off my back and hold it like a purse when I enter a museum or church and have never once been asked to put it in a locker in 35 years. J

As I traveled around eastern Germany in 2015. I found an unusual number of museums there didn't even allow my fairly ordinary (though not tiny) cross-body bag in the exhibition halls. I haven't had frequent problems in the many other European countries I've visited, though it does happen from time to time that I'm asked to visit the lockers.

Most will have locker / cloakroom options, either for a small fee (a Euro or so), free, or with lockers where you put in the Euro and get it back.

I've gotten to the point where I don't use my backpack for the day. I have a small purse (I'm a guy) for my wallet, phone, and sunnies. Idon't need anything else.

With the robustness of phones as nav aid and repository for booking info and tickets, with access to drinks and food from local merchants, etc., I see no reason to lug my backpack... I'm tired of being my own Sherpa.

And the bonus is I don't have to find a place to put it when we visit a museum with a prohibition against them.

OP you'll do a better service for yourself by checking museum websites for their current policies. There's no universal European policy on bags.

My main transportation used to be by bicycle and I've carried a bag everywhere everyday. I believe what you'll find is most museums are not allowing luggage to be brought in (checked or carry-on sizes and wheeled) because they don't have the capacity or facilities to handle such items. They're also barring folding bikes, skateboards and other large items.

One museum I visited had the policy that nothing larger than 56cm x 25cm x 45cm could be brought in -- pretty standard backpack size. When I visited, museum security were asking visitors with bags stuffed full, to put them in lockers while those with nearly empty bags and cinched compression straps were passed on by. Some museums policies state that bags no larger than "A4" may be carried inside and that other bags should be checked at their cloakroom.

My experience has been that I've never had any issue or hindrance carrying my backpack while traveling, the Rijksmuseum was the one museum where I checked my backpack at the cloakroom (I've never considered checking my bag a problem), but I've yet to visit a museum where I was not allowed to bring it in. Always ideal cause I buy too many books.