We've travelled a fair bit with Ryanair and other budget airlines with no problems. The worst flight we ever had was with one of the national airlines - Air Canada to be precise, and so bad it could have fuelled a full comedy routine - but that's a whole different story!
I like Ryanair's cabin luggage allowance because you are allowed one bag weighing up to 10 kg with maximum dimensions of 55cm x 40cm x 20cm (and they occasionally do spot checks on the weight of this one), plus 1 small bag up to 35 x 20 x 20. The interesting thing is that there is no weight restriction on the smaller bag, which means it is perfect for your jewellery, Kindle, keys, phone - all your small, heavy bits. You might already have handbags this size, but I bought a couple of bags on Amazon especially for DH and I to use on Ryanair - if you google "Ryanair small bag" on Amazon you'll get lots of options that help you make the most of the allowance.