BJTs are mostly used in common emitter, sometimes common collector configuration. I rarely see common base. When would you use a BJT in common base? For instance, what parameters are different from common emitter?
A common-base configuration is characterized as having near-unity current gain (with significant voltage gain) and a relatively low input impedance. Typical applications would include RF power amplifier (50Ω source) and the upper transistor in a cascode configuration, where it is used for isolation.
A Common Gate or Common base amplifier is perhaps the third or fourth most commonly used amplifier in the design of op-amps. You just don't necessarily recognize it as such. It plays an integral role in the design of cascode amplifiers. Wit the addition of the GG/GB amplifiers the bandwidth goes up and things are generally improved.
The most prolific use of common base is in every single op-amp (probably) ever made. Here's a very simple idea of what I'm talking about: -
With -Vin at some arbitrary DC voltage somewhere between +Vs and -Vs, the (other) signal (on +Vin) is "seen" at the emitter of both transistors. This acts as an input to the emitter of the left-hand BJT and +Vin (with or without some amount of amplification) appears on the the collector and Vout (in phase).
I'm not going to go into the different aspects of input impedance into the emitter because this is covered by the comments.