The best way to secure a stand alone flat TV on the wall is?

You can buy a thing to hang it on the wall. If you just want to secure it I suggest fishing line or twine connected to the back and secured to the wall with a hook screwed into a stud.

This is the one a relative of mine used for a 40 inch TV and it worked great. I’m sure you can find one for a larger TV or something similar.

You mean without hanging it? You’d make a bracket.
If you don’t mind hanging it flush, you can buy brackets designed for that purpose.

Check what your wall is made of.

If it’s a stud wall (timber and plasterboard), make sure you hit two different studs with wood screws, or if that’s impossible/impractical, you’ll have to use plasterboard fixings. Ideally, you’d fix one side of the bracket into a stud, and use the plasterboard fixings for the other. Make sure the fixing is good, TVs are pretty heavy things to be mounting on stud walls.

If it’s masonry, you’ll have to drill it out with a masonry bit, insert rawlplugs (wall plugs), and then you can use woodscrews to fix into the plugs. Sensible sizes would be 5.5mm masonry bit, 6–8-10 rawlplugs, and ‘inch-and-a-half-eight’ (or metric: 4.5×40) screws.

The last alternative is an adjustable bracket, as @chyna suggests. That should be fine being fixed into a stud, but downward pressure could still result in…catastrophic failure.
Just look at the potential leverage… I wouldn’t trust one at low level, especially with the tendency of drunk people to lean on literally anything. High level would be fine I suppose, unless your kids are particularly athletic/adventurous/addicted-to-adrenaline.