Got my first flat with my used/new-to-me Globe Carmel, along with an unusual-to-me experience: the tire, tube and rim strip rolled right off the rim. I rolled the bike about 10 yards to a clear area away from the road. Before I realized what had happened the rubber rim strip was tangled in the derailer (I've ordered a fabric rim strip replacement).
I'd never experienced this before with my last bike, a '70s era Motobecane Mirage, with any of the then-common clincher road tires. I typically used Continental. Got lots of flats, but the tires always stayed on the rim so I could roll the bike.
Is this common with some rim/tire combinations? This Globe Carmel is all stock, original factory components: Alex Z-1000, 700c, single wall, 32h rims; Specialized Hemisphere, 700x38c, 60TPI, wire bead, Flak Jacket tires.
A fellow at the LBS speculated the 2009 era tires, while not worn out, may have stretched the bead enough to fit loosely. I've never had tires that long on any bike so I've never experienced a tire with stretched out bead.
I also noticed the sidewalls were slightly scored by contact with the brake pad shoulders. The tapered shoulders on the brake pad seemed to have enough clearance, but I'd checked only with the bike unladen, not when I was sitting on it or with 30 lbs or more of stuff. So I've adjusted the pads lower for more clearance, and reduced the pressure from 90 psi to 80 psi (specs give a range of 75-100 psi). I'm not too worried about a sidewall blowout but do plan to replace them and keep the old tires as spares.

I'd like to replace these with the original Specialized Hemispheres, because I like the overall riding feel between pavement and very moderate trail and/or grass riding. The tires just feel good overall. But I'm a little concerned about tires that might pop off the rim unexpectedly during a flat - not a blowout, but a quick deflation over a period of several seconds or a minute.
And if I'm out in the boonies I don't want to have to lug a 30+ lb bike and gear on my shoulder if it can't be rolled on a flat, so I plan to carry at least one spare tire and a pump. This flat occurred 3 miles from home, a bit far to lug that much weight, and I didn't have a spare or pump. Fortunately a fellow in a pickup truck was going my way and dropped me at home.