Short answer: well it depends. Quarry screenings (if available in your area) are generally best for under the flagstones. Screenings are also one of the best options for inbetween the stones, but there are other options available which can be utilised to achieve different aesthetic effects. First we will adress the structural issue “what to use beneath the flagstone”.
Cement—it will crack
Sand—ants will dig it up and get it all over the place...also the sand may wash away, causing stones to settle.
Gravel—really no problems here, just use the right type of gravel. Better yet, use modified gravel for the foundation and then use quarry screenings aka grit aka quarry dust as the final leveling agent.
Cement will crack. Especially grade level cement. Especially in a climate with winters like ours in Pennsylvania. Worse way to go about it would be to lay the stone upon a bed of gravel and then cement the joints between stones. Horrible idea. The gravel base is flexible and will move ever so slightly during freeze-thaw. Well, if the base was done poorly the movement might be more then slight, but let's assume the base was done well. A gravel base will definitely move a bit—looking at any of my patios, you'd never know that, but the movement does happen. Cement is rigid—if you put a rigid top on a flexible base then systemic cracking is inevitable.
Cement is of course a fine joint filling material if the flagstone happens to rest upon a concrete foundation. But why on earth would you want to have a concrete foundation anyway? The concrete itself will crack, eventually. In a northern climate it will likely crack within ten years—and the chances of it cracking within the next three years are pretty high too. Environmental impact of concrete production is no small isue either. Anyway, I personally prefer dry stone work. More harmonious, warmer, just better.
Sand....well, if you use a really heavy sand you should be alright. Most of the sands that are packaged for sale however are way too fine. You can use a course sand for under the flagstones, sure. Back when I used to lay brick patios I would use course sand or quarry screenings inter-changably and it was fine. Them patios still look good.
Not as good as my stone patios though! Problem with using even course sand beneath flagstone is this: Bricks are of uniform thickness. Thus it's not too much trouble to just get your gravel foundation pretty close to perfect, then go ahead and screed out one quarter inch of sand for your bricks to rest upon. With flagstone however the thickness will vary too greatly—you may need half an inch of sand for one stone, but 2 inches for another—that'll cause problems. Screenings are almost the same thing as modified gravel—they are indeed one of the two ingredients in modified gravel...they are heavy enough that it really is no problem to use 2 inches on one stone and half an inch on another—ten years down the road, that patio will still look sharp.
Another reason to use screenings is because screenings also make an excellent joint filler. You do not want to use sand, even course sand between your flagstone joints because it can wash away—unless of course your flagstones are absurdly tight. For pattern-cut flagstone, yes, you can get away with using sand as the joint-filler. Just make sure the base is course sand, not fine. You will need to use fine sand for the joints however because of how tight they are. Again, ants love fine sand—but in this application, pattern-cut stones, tiny joints—fine sand will not be the end of the world—so long as the base is course, of course.
Back to screenings—when you use screenings for both the leveler and the joint filler you are creating a good scene. If there is ever any minor issue with the screenings underneath the stone it shouldn't matter too much because the joint filler will settle down and fill the void. Having screenings up top and down below, it just works out well.
You can expect to top off the screenings once within the first year—a small bit will settle or wash away. No problem, just sweep in some new material and you're good. After that, in future years, you'll be fine. My best recommendation is that clients hire me to do maybe a couple hours maintenance once a year—by no means is this nessesary, but I like my work to sparkle.
I should maybe also add that using the above system I have never once had a single patio fail. Okay, maybe one stone with some minor settling—fixable in a few minutes (and rare that that ever happens) but never any major problems. Been doing this a little while too.
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