In this case, I'd like to focus on the Napoleonic era. Were members of cavalry divisions able to wear their sabre outside of combat? Or were there regulations against something like this?
I imagine the answer changes from country to country depending on their regulations. I can speak for the Spanish Army of the time and the answer is yes and particularly for officers. You had your sabre with you if you were on duty for instance, no need for a battle, you may on duty at barracks as that day commanding officer, guard/sentinel or any other services carried out in peace time. You would also wear it in parades, festivities like Cavalry Saint Patron Day (Santiago's Day, Saint James for English speakers) and parties like balls etc.
A Spanish soldier, cavalry or not. will NEVER entry in a church armed. This required a special permissions for things like getting married or if you were a highly decorated one, few medals and titles granted the permission to enter a church with your sidearm. This is still informally observed although the rule goes back for few centuries.
Edit for few typos and structure.