So, as asked, are you having pinch flats or a simple single-hole flat?
Pinch flats are typically caused from underinflation of your tube/tire and the tube is pinched between the rim and the road hazard. They're also called "snake bite" flats.
Pinch flats look like two "pinches" or holes on the sides of the tube, right where the tube would hit the rim when the tube's flattened to the rim. The fix is inflating to what the tire calls for on its sidewall....this is indeed a case where inflating to sidewall pressure is called for.
A single hole flat, on the other hand, is just that....something like a nail, thorn, whatever, has punctured the tube in one spot and punched a hole in the tube. One problem is many times the offending object isn't really visible from the outside, esp. if it's a small thorn, and simply replacing/fixing the tube doesn't remove the problem, giving another flat almost immediately.
If that's the type of flat, make sure you remove the tire and run your fingers around the entire inside of the tire. I've caught many small thorns, wire, etc., barely visible and barely sticking up inside the tire, by running the inside of the tire after someone's complained of numerous flats and replacing/fixing tubes.