Personally, I have had one flat in the past 2.5 years.
On a Schwalbe Hurricane 26 inch tire on my 91 Bridgestone MB this week.
I prefer fast rolling tires, as light as feasible.
The Hurricane is a great front tire on a rigid mtn bike. It is pretty light and fast. Keep them up front for reliable cornering in most conditions.
This one picked up first flat after multiple rides through our Texas wildflowers on narrow, natural mulch surface trails through the meadows.
Oh well.
I have rolled Continental Travel Contacts, both 700 and 26 inch sizes(9 of them total), and don't think that I have had one flat on any of them.
And they roll very nicely in nearly all conditions excepting deep mud and ice. Great on or off pavement. Worked well enough in Ohio snow. Did go down on them on ice a couple of times, but that is to be expected.
Vittoria Voyager Hypers and Panaracer TServs and Paselas are all excellent tire choices as well, IMO.
Nice rolling tires w/ enough flat protection for me.
Haven't experienced a single flat on the Hypers on my Bianchi Volpe and these tires are starting to show significant wear.
That being said, I prefer a nice rolling tire and don't mind changing the occasional flat.
We have had 3 flats on my boys BMX bikes in the past month as they were riding a lot through the meadows in April and May, too.
We have poky plants here in ATX. August and September are the worst with our grass burrs everywhere.