what tools to use?

Scott's Cheap Flights is probably my favorite. There is a free and paid subscription; paid gets more alerts and receives them earlier. You can set your departure airports and then receive regular emails for reasonable fares on international flights. You never know what you'll get, and it does add a little adventure to your inbox. I got roundtrip fares from Indy to Madrid for about $350 using this service.

I have a limited airport, but I like Matrix ITA for checking pricing over a month. It allows for airports within a range that you determine; however it tends to want a destination.
Skyscanner may be your better option - for searches, at any rate. You can enter nearby airports, click on the destination bar and a dropdown allows you to chose 'everywhere', and you can look at a date range.
Also, determine what airlines you prefer and sign up for the sales alerts.

Jack's Flight Club sounds like it's a similar concept to Scott's but based in the UK. Whilst many of the flight deals are long haul from the UK every Friday they publish a list of deals for flights within Europe for a long weekend. It is, however, a paid subscription (£35 pa I believe).

Manchester Airport is better connected but also Leeds Bradford is close to Sheffield. Ryanair operate a number of cheap flights from Leeds Bradford and your only way to get the cheapest flights is to book direct with them. There's no reason to use third party sellers for cheap flights within Europe as you're unlikely to get a better deal and it also leaves you without much recourse if something goes wrong whereas booking direct with the airline means that they have a duty to get to your destination.

I think there must be several very workable airfare search engines that would help you. I suspect it's more a matter of fiddling around with one of them to see all that it can do.

My needs are different from yours in that I start out knowing my arrival and departure airports in Europe and am not willing to travel far from home in the US to save a bit of money. I use and like Google Flights. Go to the intro screen and select a departure city (note that "Washington DC gets you IAD, DCA and BWI) and click on "Explore destinations" in small blue type at the right below the input boxes. On the next screen type Europe in the destination box. Click on the small blue calendar icon near the upper left and choose a month (within the next six months) and a duration (weekend, 1 week, 2 weeks). You'll see a map with airports marked by blue dots, some with price bubbles. As you zoom in, more price bubbles will appear in the area covered.

Obviously, this tool is somewhat limited. It only works with round-trips, for one thing. If you're looking for multi-city (open-jaw) possibilities, you can try pairing two cities with inexpensive round-trip fares to see how the multi-city itinerary prices out.

Skyscanner.com is the usual recommendation for researching intra-European flights. There may be other, equally good, options.

May not apply to you, but having been part of a FF program through an airline, I get notification of flash sales at attractive prices or miles. Trick is that notice is short, 1-2 months out, covers only specific dates within which to depart and return, and best deals from specific airports to specific airports (limited routes). The deals also represent more off-peak or off-season travel to the destination.

I use Delta and get notice about once a month, not always Europe, and since I am not yet retired, often cannot take advantage, but though when that day comes might be an option. I have over the years hit one or two of them though.

We now use Google Flights and have found lots of good deals. We start all of our trips to Europe by taking a daytime flight out of EWR, JFK or BOS.

It's my understanding that the OP is looking for suggestions for tools/sites to use to search for flights from airports close to Sheffield in the UK to European destinations as opposed to seeking cheap flights from the US to Europe. In which case, whilst much of the aadvice given is good advice it's not entirely relevant to the OP's question to which the most suitable advice is to search on the budget airline's sites as that is where you'll find the cheapest flights and also the added protection.

With respect to JC's post; I don't think that I need any help getting to/from Sheffield to any place in Europe. My belief is that once I have gotten to Europe the short haul trip will be simple and relatively inexpensive to schedule.

The problem that I am finding interesting is:

How do I identify really good flight from any of 5-7 airports in the US to any of 10 or more airports in Europe on any day in any given month? I haven't even begun to think about open-jaw options. Of course, I could just strap in and perform an exhaustive search. I was just wondering if any of you had any good ideas.

Thanks.

matrix.itasoftware.com

www.skyscanner.com

www.cheapoair.com

My basic tools. Skyscanner turns up budget flights in Europe that don't appear on North American-oriented search sites. The Matrix comes from Google as does Google Flights but in my unscientific experience turns up more choices than its cousin (but won't book them for you). Cheapoair is easy to use and I like its information but only book through them on itineraries with two or more flights where the airlines don't sell each other's tickets.

https://www.vayama.com/ is another agency that gets decent reviews but I have no personal experience with their service.

Seek out a search engine website, such as Kayak, and create multiple options for flights with each of your airports and receive daily notifications on prices. That way you can see what comes back.
With that said, are all these airports truly equidistant for you and not just measuring mileage, but the time considering the traffic to get to them? I live in an area where DCA and IAD are most convenient for me, and although BWI would technically be a 45 minute drive, getting to the airport for an evening flight (as they most are to Europe) would add about another hour and lots of stress about getting to the airport on time, and as such, not worth the $100-$150 I may save (I am very close to DCA and IAD has dedicated lanes so it is super easy to get to). The eastern corridor where most of the airports are located are burdened with a lot of traffic! Also, the quickness of getting to IAD usually means I do not have to take time off work the day of my flight, but I suppose that would not be an issue if one is retired. I say all this only because "value" may be defined in different ways.

I'm with Amy -- Scott's Cheap Flights fills my inbox with tempting opportunities from multiple airports to multiple destinations. If there's a good sale, I check on the airline site to do my booking.

I've read some very negative things about CheapoAir and suggest that you do careful checking of reviews before using it. I also have a vague recollection of red flags related to Vayama, but I'm less certain of that one.

You can just leave the destination box empty with Google Flights and it returns the map with prices labeled on possible destinations. And while you have to search date by date, Tuesday and Wednesday are generally lighter travel days with slightly better fares, and a Saturday night stay is still often required to get the cheap fare buckets and it's not that much playing around to get different dates.

You can still learn about some good deals ahead of the bloggers by checking the 'Mileage Run' section of the FlyerTalk message boards.

To me it sounds like Scott’s (for US) and Jack’s (for UK) would be the most useful for you.

I use Skyscanner to search for intra-European budget fares.

I’m a very long time user of matrix.itasoftware.com—long before Google bought it and changed its format. It used to have the word Beta in its verbiage. I love that I can pick a reliable airport then choose two more nearby (Search Nearby) airports within a 75-500 mile radius on both ends of the search. I can quickly check a Month Long calendar rather than Specific Dates. I can even do multi city flights from https://matrix.itasoftware.com/. It holds some destinations you’ve previously searched in its memory so you can quickly repeat your searches weekly or monthly. We’ve gotten some terrific and unexpected airfares using this website then booking with the airline for many years. From the west coast our goal has become airfares for around $500 RT rather than $1000 RT/per person. Have fun with your new retirement and exploring possibilities.

EDIT: my husband noticed that itasoftware now allows you to enter more than 3 airports into your search, a good feature.