Personally I know what I'm willing to pay, I follow United's pricing from Dulles and I'm OK having the belief that I think I have some kind of grasp of that movement.
And that means flying on off-peak days: Tuesdays, Wednesdays, and Saturdays are often much cheaper days to fly
I can be pretty flexible, I do shop by days, and looking back at my last 8 ticket purchases 3 where for flights on a Wednesday, 2 on Tuesday, 2 on Sunday, and 1 on Saturday. Is there truth to that statement? I don't know, I do find myself looking at flights on Wednesdays first, because I've purchased the lowest priced Premium Plus tickets for Wednesday flights, but I always scroll through the rest of the week and check all days.
When one checks prices, it may be the case that Wednesday is the cheapest day to fly on, but there are too many other market factors that drive price so that the so called cheapest day to travel changes. There's the tried and true supply and demand. Premium cabins vs. Economy cabins. Are we talking business routes or leasure routes or routes with a mix of both. Is it a seasonal route? What are the arrival or departure airports?
There are too many variables to make "flying on off-peak days...much cheaper days to fly" some sort of maxim that I would want to live by.