Are plain diodes uncommon?
"Plain" diodes (silicon diodes, germanium diodes, etc) are very common. As Gerben mentions in his rather good analogy, with Zener diodes you have a wider range of options. The actual number of discrete options for Zener is greater than that of silicon due to the wide range of voltages. However, the quantity of silicon diodes on the market and in use in the world is far greater - you just don't have such a wide choice, since there's no need for it. 

Schottky diodes, too, give a greater selection due to them having more parameters of interest (switching speed, leakage current, forward voltage, etc) that you don't really care about with a boring silicon diode. 

To put it another way and use another analogy - imagine walking into a builder's merchant's. There you have piles of similar items:


Bricks
Breeze blocks
Concrete Slabs


Bricks there is only maybe one or two different types. However there are huge piles of those couple of options. People use a lot of them, but there's not much to distinguish one brick from another.

Breeze blocks there may be a number of different sizes, some hollow, some solid, etc. There's less of each type, but there's more piles.

Finally there's the concrete slabs. You have different colours. Different patterns. Different sizes. In short, there's many many many different options. So there's many many different piles. However there's only maybe one pallet of each, compared to twenty pallets of one kind of brick.

So although it may look like there are more Zener diodes around, all there actually is is greater choice of parameters for Zener diodes. There is actually far more silicon diodes around in the world. You'll be sat within a few feet of tens or hundreds of silicon diodes right now, but maybe just a small handful of Zener diodes.