In the USA it matters which day you fly. Would this apply St. Louis to London? Plan to fly Sat. to Sat. (2 weeks) in late April but am flexible: a Mon, Fri or Sun would work. So far, I'm sensing it doesn't make a difference with International flights.
Be flexible and play around with different variables (including dates/days of the week, airlines, itineraries, and even airports) to see the effect on any itinerary - you'll get a variety of prices and will have to make trade-offs if saving money is the primary goal.
Monday flights might be slightly cheaper compared to Friday, Saturday or Sunday, but probably only by $30 or so. Saturday or Monday might be better than Friday or Sunday with regard to crowds at the airport.
Connections and layover times are more critical, since you don't have direct flights from St. Louis.
If you look at an airline that shows prices day by day for a month, you will see that some days are more expensive than others. I have found this on British Airways and Condor, to name two. It does not seem to be consistent for each day of the week, but more random (based on how the plane is filling up).
Looking at Condor in late April, the fares from Seattle to Frankfurt are considerably more expensive on Wednesday and Saturday than on Monday, Tuesday, Thursday,, or Friday. $829-849 for Wed. and Sat., $459-539 the other weekdays. Sunday is intermediate at around $659. But it changes in May, with less variation between the days.
Of course, other airlines will be different.
To me the best way to tell is to go to my airline of choice, put in flexible dates and then my experience on the Delta site is that a calendar pops up with prices on different departure dates.
It does often make a difference so look carefully.
St Louis is my usual departure point. Prices on American for April are looking around $1100 round trip from STL to LHR. Which is not a terrible price, but if you want to save money (depending on how many tickets you are purchasing) you may want to look at leaving out of Chicago. There are dates in April around $450 - $600 return on AA. Depending on your location, it may be worth the drive. Most definitely play around with dates and airports. Those connecting flights into St. Louis can really be pricey brought on one itinerary. I was looking at July from STL to LHR at $1700 - $1800. I was not willing to pay this price. For this year airline prices out of STL seem more expensive than usual . (I ended up using airline points).
I checked Brigitsh Airways for STL to London roundtrip in April. Their website shows lowest price for a week centered around your chosen date so you can compare. But beware: the low price they show on the calendar is a “hand baggage only” fare; no checked baggage (and maybe other restrictions). So looking at April 14-28, the price differences on sees on the outbound flight ($639 versus $689) is illusion because that $689 is on a day when the HBO fare is not available.
The biggest fare difference is between early April and late April. If you can go April 10 (a Wednesday) or earlier, you can get the outbound flight for $320 (HBO) or $375 in regular economy (with checked bags). All outbound flights starting April 11 or later were in the $639-$689 range.
This brings the fare for a two-week trip starting 10 April or earlier (like Monday, 8 April) down to $923 for regular economy, or $813 for the lower hand baggage only fare.
Some airlines have "low fare calenders" identifying price by calender day. Examples:
Select "Low price calender" when doing a fare seach.
When doing a fare search, the Alaska web page shows price 3 days each side of prefered date.
When you travel varies more than the day of travel in my experience. Although there is business traffic around weekends, many tourists are flexible about when they fly (other than the Germans who all seem to travel on a Saturday!). Travelling during major school holidays such as summer and Easter pushes the prices up. Flying the last week of April maybe cheaper than the middle 2 weeks for this reason.
Europeans have 5-6 weeks paid holiday a year, so are less precious about squeezing every last day out of their trip and are therefore more flexible about when they fly.
GoogleFlights also has the fare by day so you can compare the most economical dates to fly.
I've made the mistake of booking a hotel and later the flight, when it probably should be the other way around.