There are hundreds, if not thousands, of optical designs.
The four listed above are some of the most common and well known
for mirrorbased systems. 

A spherical figure is questionably the easiest to
produce. Next is the slightly deeper and therefore more difficult
to produce elliptical figure. Deeper still is a parabolic and
even further still is the varying degrees of hyperbolic. A sphere
has a Kconstant of 0/zero. A parabolic has a Kconstant of 1.
Anything between a 0 and 1 is some form of an elliptical shape.
Anything past a 1 (i.e.: 1.25) is a hyperbolic. 

A f3 primary mirror has a steeper curve than a f4.
This is true no matter what the desired figure: sphere, elliptical,
parabolic or hyperbolic. One of the hardest mirrors to figure
is a fast hyperbolic. Some cutting edge applications have called
for focal ratios as fast as f0.85. Most large, 3 meter and larger,
telescopes in the past five years have used primary mirrors of
f1.7 to f1.25. 

Typically the focal ratio on most commercially available,
under 1 meter, Cassegrains is f3. 

Most convex surfaces are expensive and therefore difficult
to test. The exception being the spherical convex as used by
the DallKirkham design. This design in particular uses a primary
mirror that is only slightly more work to produce than a sphere,
yet less work than a parabolic. The secondary on the DallKirkham
is also much more feasible to produce and test. Of the three
Cassegrain types listed above though, the DallKirkham
has by far the worst offaxis image quality. It can be an
inexpensive, specific purpose onaxis instrument. Both the Classical
Cassegrain and the RitcheyChretien Cassegrain offer a more diverse
instrument with better inherent optical quality across a field. 

Any four of these designs can be built from the ground
up with a corrector in mind. The corrector
can greatly enhance the optical quality of the instrument at
the focal plane/eyepiece. Correctors have and will continue to
be utilized on optical systems. There use will continue to grow
as more become aware of the dedicated corrector's potential.
Even the refractor has not been isolated from the correctors
use. Both in add on correctors for refractors but also inherent
correctors that have become part of the optical system, i.e.:
doublets of the previous century being replaced by triplet designs.
The third lens element is arguably a corrector that was blended
into the design. 

