Having used them as a bed for a year, there's airbeds and there's airbeds. The cheap vinyl ones puncture if you so much as look at them. The heavy, rubberised cotton ones are a lot more durable, slightly less comfy, but do tend to die of old age, especially around the valve.
Self-inflating mats tend to me made from decently durable material, so are somewhere in the same category, punture wise, as the good sturdy airbeds. They also have the distinct advantage of having more than just air in them: Common misconception with airbeds is that the air will act as an insulator, but in reality, convection currents mean they conduct heat away from you surprisingly well. Self-inflating mats have some kind of foam (which is what makes them self-inflate), which prevents the air from moving around, making them much warmer for their thickness. They're also less faff than airbed pumps, though people with hips and shoulders may find them significantly less comfortable.
Exped mats (basically high-tech airbeds with an insulating filling) tend to be the gold standard, and meet that elusive combination of being actually comfortable and reasonably lightweight. I don't think they're immune to punctures, though, and you do have to perform comedy CPR to inflate them.
If you're not too fussed about weight/bulk, I recommend the thicker Alpkit self-inflating mats.
As for what's causing the punctures, if not something like sleeping bag zips, think about how you're transporting/storing them?
