Maximum hand luggage volume?

22" x 14" x 9" is used by American Airlines, Delta, United, Virgin Atlantic and probably quite a few more.

Ryanair is 55 cm x 40 cm x 20 cm that is 21.5 x 15.7 x 7.8 Inches. There are other low costs using this size. Note that many airlines use 55 x 40 x 23 but according to https://travel.stackexchange.com/a/30181/4188 the sizer is only 21.2cm so I would not expect a 23cm bag to work out with Ryanair. Of course unless you have a hard shell or packed your bag to the gills some squeezing might allow for a 23-thick-advertised bag to fit the sizer.

Therefore 21.5" x 14" x 7.8" / 55 cm x 35 cm x 20 cm is the biggest bag you can safely carry. This is the IATA recommendation as well per this 2015 June BBC article. The closest non-wheeled (because that's just extra weight which you don't need when you have a severe weight limit often anyways) I am aware of is the Magma Digi Control XL backpack is 54 x 35 x 20 cm which I just bought exactly because of this and also because I have a 19.5" monitor to carry on and it happens to fit into its laptop pocket which is pretty rare and it has a second laptop pocket which pushes it into unicorn territory. Has its own drawbacks: it doesn't open like a suitcase or a "convertible" backpack and also there is no size handle.

You might be able to get away with slightly bigger: the main US airlines' sizers are actually 23" x 15" x 10" United Delta so we can conclude that 15" instead of 14" is doable which increases your maximum bag to 21.5" x 15 " x 7.8" / 55cm x 38cm x 20cm. This is the size Emirates actually allows.

So if you want to go nuts, buy a 22 x 15 x 9 / 56cm x 38cm x 23cm duffel style bag if you can find one, pack lightly (8kg is not unheard of as a limit, 10kg is common) and prepare to squeeze. Perhaps you want the Red Oxx Medium Aviator Bag and a few of their luggage compression straps that you set up to cinch it to a width of 9" -- the other two sizes are just what you need, I have the Extra Small One, I find the price surprisingly low given the building quality. Of course, there's no structure but that's what you want from it. Get packing cubes :)

You definitely want to check out the Carry On Guy.

Ryan Air tends to have one of the smallest (by dimensions) carry-on luggage allowances and also have one of the strictest. 55x40x20cm (21.5 x 15.7 x 7.8 Inches) But they allow a generous weight of 10kg. The most restrictive weight allowance for a carry on seems to be about 5kg.

You can be pretty safe if you get a light bag that meets these dimensions. However another common dimension limit you will find is: 56x35x22cm (22 x 14 x 9 Inches)

This means to be absolutely sure, your bag should be at the most 55x35x20cm

EDIT:

To throw an extra spanner in the works. Wizz Air have a more restrictive "free carry on" luggage allowance, while charging a fee for the "Large cabin bag".

Small cabin bag (Free): 42x32x25cm

Large cabin bag: 56x45x25cm. - Online Price: €10 - €18 depending on route, Airport Price: €40

We have taken multiple airlines throughout Europe during 28-40 day trips over the past 2 summers. I did a lot of research and used the 55cm x 20cm x 14cm Cabin Max Lyon Wheeled Backpack Convertible carry-on packed to a max of 10kg (combined weight with my small personal item--in my case, my purse in my son's case, his laptop bag). It was perfect for all flights. One of the airlines had a max weight of 8kg so I "sacrificed" our aqua shoes and a top and found I didn't need to do so because no one even looked twice--but better safe than sorry. Just bought 2 new ones from Amazon for trip this summer. They are just about $70 each. The fabric did start to tear a bit on the bottom at end of last trip but still well worth the price for all the hassle we saved not checking baggage and not paying oversize fees. Happy Travels!