Common emitter amplifiers provide excellent gain but fairly low input impedance and some distortion.
It is possible to trade off some of the gain for better input impedance and lower distortion by introducing an unbypassed emitter resistor.
At high frequencies they also experience Miller Effect where the base-emitter capacitance is magnified by the gain of the transistor and appears as a large capacitance across the input of the amplifier. This can severely limit the high frequency response of the amplifier.
Common collector circuits ( also called emitter followers) are mainly used as buffers. They provide good isolation between input and output. They have a high input impedance and low distortion and good frequency response.
However, they have a voltage gain of less than 1, so you get less voltage out than you put in.
The voltage you do get out is at very low impedance, though, and this can be an advantage if the amplifier is being used as the output device of an instrument.
Common base amplifiers are rarely used now. They give high gain and good frequency response, but they have a very low input impedance which makes them dofficult to drive. Their input is almost like a short circuit.
There are some oscillator circuits that use a grounded or bypassed base and these work very well at UHF.