We might be able to share some ideas that have worked for us - or just times we were lucky.
I'll start:
Versailles on a Saturday around the first of October on a fountain day. It wasn't crowded at all. I'm guessing rain in the forecast kept the locals home. The palace and gardens were very uncrowded, even though the rain never materialized.
The Louvre in mid-September, can't recall the day (but I usually avoid the most busy days). I had a museum pass in hand and arrived a little earlier than planned, about 40 minutes before opening. I had no bag and was first in the pass line. When the Louvre opened, I was waived through security, was the first visitor that day and walked straight to the Mona Lisa. I had the room to myself and a Canadian couple who caught up to me (they were first in the prepurchased tickets line) for a good ten minutes before anyone else showed up.
We beat the line at Neuschwanstein by arriving well before opening (mid October). There were maybe 25 people ahead of us and we booked our tickets quickly for early admissions to the two castles. When we were finished and ready to leave, the ticket line was worse than Disneyland.
There are more that follow the general theme of arrive before opening for the busiest sights - Notre Dame towers, Concierge.
Mt. St. Michel (mid September) was the opposite. I arrived later in the afternoon, walked straight up the ramparts to the Abbey (an hour or so before they closed) visited the Abbey then strolled back down the hill. There were crowds on the main street but not to the point of being impassable.