I was challenged by a warden in a National Trust property when she saw my camera, & told that photography wasn't allowed. I bet that lots of people take photos & get away with it, esp with phones etc! I tried to interrogate her about why (politely) but she didn't know - it was obviously just a policy handed down from above.
I can't for the life of me think why they have that policy, and for the NT it's a negative in marketing terms (deters me from visiting their places).
Anyway, it varies a lot. In a quite different place, under another organisation, my camera didn't cause alarm, but when I nipped back in with a tripod over my shoulder I was called to the desk and asked to sign a form stating that any photos wouldn't be for commercial use. This I understood (sort of).
For other places, I've phoned in advance about taking photos and/or using a tripod, and it's been fine. I also have a guerrilla mode of just getting on with it until possibly challenged.
Yeah I was thinking about buying a monopod actually....for my next trip. Does it really stabilise well?
I'd say from experience that a monopod's worth 1 or 2 stops of exposure-conferred sharpness, if that means anything. In other words, less than what a modern lens with vibration control contributes.