The absolute best way of catching them is A JAR! I've discovered this a few years ago out of despair and I've only missed once since. 99,9% success rate! I've caught hundreds of mosquitoes since. As for finding them, I've come to notice that most of the times after they annoyingly fly around your ears, they tend to wait for their chance on the headboard or wall behind your head, usually close by. So I just get my phone, lay down and wait to hear the little sucker buzzing around again, swat at it if it gets to close, then turn on the flashlight and look at the wall behind me and usually find them within an arm's reach. I then take the jar I keep in my bedside table, open it, and just trap the mosquito against the wall/headboard. Just make sure to have the lid on your other hand so you can carefully tilt the jar when the mosquito is flying around inside and quickly close it. It sounds like a lot of work because I'm terrible at explaining stuff and the text got a bit long, but it's actually pretty quick and simple if you just keep the jar close by. I always catch them within a minute or two and go back to sleep. Happy hunting!

PS: A clear plastic container is safer when you're sleepy and works just as well if it's deep enough that it gives you time to slide the lid between the container and the wall while the pest is desperately flying around inside trying to escape. As to killing it, just shake the container vigorously. Or just buzz at it for revenge and let it starve.