My wife and I are going to be flying London Stansted to Dublin. We are only bringing a carry on bag which will definitely fit the overhead but I am not sure if it will be under 10kg (roughly 22lbs) Has anyone ever dealt with Ryanair checking the weight or is it usually the dimensions they care about?

You can never say never with FR, but it’s been more than 10 years since I’ve had my carry-on weighed at the gate. Provided your bag doesn’t look to big or heavy then you should be okay.

Unfortunately you never know as it experiences will vary per location / per day / per gate, etc...
Flew once last year out of Stansted on Ryan Air and they did not weigh carry-ons.

As a general rule, if your bag does not look close to the size limit it won't be weighed. Usually backpacks are given less scrutiny than wheeled bags.
I travel with heavy fragile photography gear so have an overweight carry-on (backpack) on every single flight I fly on and never been weighed.
BUT my advise would be regardless of weather outside is to wear a jacket with pockets. If you witness people ahead of you getting their carry-ons weighed move the heaviest items to your jacket pockets. You can then decide to move them back into bag later if you want.

Most likely your backpacks will not be weight checked, unless it is apparent to gate staff that they are too heavy/that they may contain much more than 10 kilo.

Yes, I agree with others on this thread. Provided the bag does not look too big you should be fine. It's rare that I have seen staff weigh anything!

I've had circumstance where the gate staff get very attentive about baggage size when the flight is fully booked. Multiple announcements about allowing free storage in the hold for volunteers. Of course few enough of passengers take that offer up so then the gate staff start weighing and sizing any bags that look too big or too heavy.