With a full-wave bridge rectifier you have four diodes, resulting in two diode drops, and the capacitor value is roughly half what you need for a half-wave rectifier with the same ripple. If you have a center-tapped transformer you can use two diodes to get full-wave rectification with only a single diode drop, but the extra transformer tap and associated connections are not free. For rectifying mains voltage or voltage from a typical AC-output wall plug adapter you don't have the option of a center tap (usually).
The four-diode bridge uses a transformer capacity most efficiently, center-tapped (two diodes) full-wave is better than half-wave but worse than the four diode bridge for voltages much higher than a diode drop.
With a half-wave rectifier you have only one diode, and only one diode drop, but you need a larger capacitance (typically physically about twice the volume). It's the least efficient way to use a transformer (as a single rail anyway).
Most switching power supplies use full-wave rectification on the input. On the output they may use half or full wave as the higher frequency means the capacitor size is not as important. With a flyback converter you don't have a choice- it has to be half-wave.
Half wave rectification is sometimes used with an AC-output wall plug adapter to produce balanced +/- rails for audio. I have a hydrogen-oxygen electrolysis welder (US made ca. 1968) that uses half-wave rectification probably because the optimum voltage for the gas production cell is only a couple volts and two fat stud-mount silicon rectifier drops would be less efficient and more costly even though the current is many amperes (no filtering required in that application). These days we'd probably use a switching supply with Schottky diodes at the output.