About the only one I could say is a "given" would be the Borghese Gallery in Rome, where you have to check nearly everything--coats might be the only exception. Otherwise, it completely depends on where you are. I carried a large purse everywhere except the Borghese and Uffizi, and I think I would have been allowed to keep it at the latter, but we decided to check the bag for comfort's sake. I tried to check it at the Capitoline Museum and they told me to keep it!
Fanny packs will probably be OK at museums that allow large purses, but it is just going to vary. The good news is that the baggage checks I saw were all simple, secured, and free.