For example, I want to travel from London to New York. My dates are flexible departure (July) and return (September).
I want my output to be the dates for which I'll get the cheapest flight (instead of manually trying out multiple options on Skyscanner or any other flight search website)
This seems to be a pretty common feature. For Expedia, for instance, you can choose "Show flexible dates". Google Flights also has it under "Date Grid". It doesn't seem to show time of day, however, so searching for the lowest fare will likely get you a lot of red-eye and such.
Beat That Flight (my site), has this feature which I term Flexisearch, eg:
For Australian based users, at least, it also does this focusing on popular routes.
Example: Melbourne to Tokyo:
As other answers have mentioned, there are numerous sites offering this. However, all those are showing you which is the cheapest day to buy a flight now. This allows you to find a low price but those may not be the lowest prices because it is highly likely that purchasing the same flight a different day comes to a different price.
Instead of doing the search now, you can have a service searching for you. These are called Flight Alerts or Travel Alerts. You subscribe to a service and set some parameters, such as origin, destination (can be specific or broad) and frequency of alerts. You can also allow them to select the cheapest dates or give constraints such as weekends, months, etc (depending on the service).
If you are keen on getting a certain flight, you can ask for daily alerts and every day you will get an email that lists the cheapest flights to your selected destination. You can choose weekly alerts to diminish volume of email but you might miss the very best price in that case. The main service I have used for this is Kayak and that in fact purchased from the alert email several times. There are a lot of flights to discard though since you cannot constrain the duration of the trip or just because some might be overly complex for your tastes.
Thgere is an app called Azuon which I used for a long time. It was far more flexible than any search engine. It required a subscription for full features, but even in demo mode, it would give you prices and dates, and then you could use another service to find what airline it was on that date.
I stopped using it because it started giving absurd results, but that was on Mac with a windows emulator. It might be that the unwrapped app on Windows still works.
