Are there any child-free flight options?

Children and infants are not allowed in the emergency exit rows of airplanes. However, it only provides a guarantee that a child won’t sit next (laterally) to you — one might still be behind or ahead of you in a non-exit row. If you can find a plane where the exit rows correspond with a change in class or there are multiple exit rows, you might be able to build more of a buffer. I often use seatguru to find the row directly ahead of the exit row because there will be no children behind me - or the row behind the exit row if I need to do work on my laptop, because exit row seats are not capable of reclining.

In addition personally, I find that earplugs or a good set of noise cancelling headphones along with good music or a movie of my choice more than drowns out the world. If you’re really bothered, in-ear earplugs combined with over the ear noise cancelling headsets (or in-ear noise cancelling headphones combined with over the ear ear protection) will give you around 40dB of noise reduction - more than needed for most children.

Finally, parents of small children often can’t afford to fly business or first class. There’s also greater seat separation and/or privacy partitioning in those classes. You may find that one way that airlines offer a “choice” (albeit expensive one) to business travelers who absolutely need to get some sleep or work done.

Your only real option is to charter your very own aircraft and crew (unless you're a pilot yourself of course, in which case you might be able to rent an aircraft and fly it yourself if your ratings allow you to fly an aircraft with sufficient range and passenger capacity for your purpose).

The cost of that is pretty steep, several hundred to many thousands of dollars PER HOUR just for the aircraft, not taking into consideration payment for the crew, fuel, landing fees, etc. etc.

It's a great way to travel, but it's expensive. Maybe some day I'll be allowed and able to have my own aircraft, but for now I'm just a student pilot learning the ropes.

I don't know about kids of age, but when I was flying with infant I have noticed that places for parents with infant are all in the forward section. So you can try choosing seats in the back. This was just one airline though, so your mileage may vary.

Come to think of it, back section is often considered undesirable, so you could choose seats in the back to minimize the absolute number of neighbours around you - provided that flight isn't packed.

Then again, by choosing flights which are rarely full (this probably means more expensive airlines on less popular routes), and there choosing undesirable seats, you could minimize the amount of kids around you.

In my experience over 30 years of air travel, the only solution I have discovered to simulate a child-free flight is to plug in and turn on my Bose noise reducing headphones. It's not perfect, but if you crank up your favorite music, the headphones can lull you into a Thoreauly Walden-Pond state of solitude. It is also effective for ignoring your rude adult row mates.

Can I request a child-free option on a regular flight to have no kids sitting nearby?

Perhaps not from the airline, but you could always see if other passengers in seats you feel are more optimal are willing to switch.

Personally, I love kids, even before having my own I've always loved the energy and raw emotion they show. I also love parents, doing the best they can to bring up their young ones when they really have no idea what they are doing.

With this in mind, sitting next to screaming kids on a plane has never bothered me. If I'm feeling social, and the parent welcomes it, I'll easily try and help out stop the screaming by just pulling funny faces, playing peekaboo, lending them my tablet and that sort of stuff. If I don't feel social I'll fall asleep or put in my head phones and ignore it.

the purpose of the question is about trading money (maybe time) to comfort – fixerIt (comment)

Then you are in luck, because as well as loving kids and parents I also love money. Even if I'm not feeling social and don't want to switch seats the added incentive of some money for doing so would almost guarantee it.

How much does it cost to have a child-free flight compared to a normal one?

£50 would guarantee it from me but other passengers may do it for less. You could always haggle as well, depending on my mood I'd probably go as low as £20.

Of course some people may take umbrage at you trying to get out of sitting next to a screaming child and think you a bit conceited for doing so and suggesting they should sit there instead. To try avoid hurt feelings as soon as you ask I'd suggest leading with a plausible reason as to why you want to switch (even if it is a white lie). Something like 'I've a meeting as soon as I get off the flight and could really do with getting some rest.' rather than just 'I don't like kids.'

Something that makes the person you want to switch with feel like they are helping you out will make them more likely to help.