Do the SPI modes have a specific application?
I don't think there's any preference. SPI bus never has been formally standardised and it's been around almost forty years. So pretty much every combination of these "modes" have been used by different vendors. Don't think the mode numbers always mean the same thing either. 

https://www.byteparadigm.com/applications/introduction-to-i2c-and-spi-protocols/

To make things more interesting there are "nonstandard" implementations, one popular choice is adding a "ready" signal to the slave. SPI was originally designed (in 1979!) to communicate with simple devices which would have guaranteed response times. This can cause problems with more complex devices such as auxiliary microcontrollers or, say, standalone communication modules. 

Throw in dual spi (can run in half-duplex mode with 2 bits transferred simultaneously) or quad spi which adds two extra data pins. There are, naturally, different incompatible versions of both. 

So that's why you have these different modes, it's because there's no standard and you have to support different behaviors to maximize compatibility.