No simple answer exists, because to much parameters are involved.

First it  depends on the active medium inside. In most cases the active medium has a thermally induced lens (e.g. lamp or diode pumped rods) and this lens has to be taken into account.

Have a look on the resonator stability diagram ( Hodgson, Optical Resonators and Beam propagation, Springer NY 2005 or Siegman, Lasers, University Science Books, Mill Valey). You will realize that the resonator types: plane-plane ,concentric , confocal, hemi-shperical are located on the boundary stable-unstable and can be used only, if the system is stabilized by the active medium.  

The concave-convex resonator has the advantage of high mode volume and high extraction efficiency but the disadvantage is that the position of the beam waist is sensitive against distortions of the active medium. It makes problems with the beam handling.

In most cases a plane spherical resonator is used, where the waist is located on the plane mirror.

If the laser is a high gain system (CO2 or Nd-YAG) unstable resonators are used with a high efficiency.

The type of resonator depends on the actice medium, its gain, the environment ( thermal and mechanical sensitivity, misalignment sensitivity) and the application of the laser beam (material processing, interferometry, metrology). Look into the Hodgson book, where all thes problems are discussed in detail.

Oh, now lasers are of high cost and rarely break down, and I did not examined their interior for a long time....PLANE is acceptable if laser medium has gain. Low gain requires spherical mirrors to better eliminate diffraction losses. Multi-element laser must have optimized resonator that is not exactly confocal or concentric, it is intermediate to better fit apertures of all elements....

My guess would be that plane-parallel is the most frequently used, because most of the semiconductor lasers have cavity made by the crystal faces, and laser LED is likely the most common type of lasers now.

No simple answer exists, because to much parameters are involved.

First it  depends on the active medium inside. In most cases the active medium has a thermally induced lens (e.g. lamp or diode pumped rods) and this lens has to be taken into account.

Have a look on the resonator stability diagram ( Hodgson, Optical Resonators and Beam propagation, Springer NY 2005 or Siegman, Lasers, University Science Books, Mill Valey). You will realize that the resonator types: plane-plane ,concentric , confocal, hemi-shperical are located on the boundary stable-unstable and can be used only, if the system is stabilized by the active medium.  

The concave-convex resonator has the advantage of high mode volume and high extraction efficiency but the disadvantage is that the position of the beam waist is sensitive against distortions of the active medium. It makes problems with the beam handling.

In most cases a plane spherical resonator is used, where the waist is located on the plane mirror.

If the laser is a high gain system (CO2 or Nd-YAG) unstable resonators are used with a high efficiency.

The type of resonator depends on the actice medium, its gain, the environment ( thermal and mechanical sensitivity, misalignment sensitivity) and the application of the laser beam (material processing, interferometry, metrology). Look into the Hodgson book, where all thes problems are discussed in detail.

For a discussion of applicability of different types of resonating cavity for lasers, I'd like to mention that the wikipedia page, which contains the Fig. of the suggested cavity schemes, does not even mention some possibilities, e.g. ring cavity lasers, or fiber-optic lasers. Probably, superluminescence should be also mentioned here, since boundary between laser LEDs and superluminescent LEDs is difficult to draw. Those types have there own advantages for specific applications.

Long range application, Remote sensing but Low beam quality: Unstable resonator

Short range application, High beam quality or Single mode operation etc : Stable Resonator

Note that in Stable resonator ,  the wavefront   and the mirror surface should match to achieve high beam quality.

The application dictates which type of resonator is most efficient. It is a compromise between technology and art!

By comparison with modern fast-flow lasers( up to 25 kW), the slab laser is extremely compact. In these lasers, which are available with powers from 1 to 8 kW, an RF gas discharge takes place between two copper electrodes with large surface areas. The small gap between the electrodes allows maximum heat dissipation from the discharge cavity via the water-cooled electrodes, producing a comparatively high power density. The unstable resonator uses cylindrical mirrors and produces a highly focusable beam. In external, water-cooled, reflective beam-shaping components, a rectangular beam is transformed into a rotationally symmetrical beam with  a better beam quality.

Unstable resonator extracts higher energy from gain medium at the expense of beam quality (larger M2).