What is the best display type?
There is really no such thing as the "best" type of display. Rather, we can think of the best type of display relative to some standard.
First, let's consider the 3 types of displays.
Advantages: These displays can be very sharp and very bright.
Disadvantages: They are limited in screen size and can be very heavy.
Advantages: Rear projection units seek to offer the convenience of a single appliance without the limitation on size that direct view sets impose. Rear projectors are also bright enough to watch in a well-lit room.
Disadvantages: Both because of the fact that they must use a mirror to keep the projection path compact and because of the rectilinear screens they use to optimize brightness, rear projection images (of all display technologies) are less sharp than either direct view or front projection units.
Advantages: The most obvious advantage of front projection is size. 100" images are typical of front projection applications. The projectors are also quite small and light weight. Furthermore, even including the cost of the screen, many high-quality digital front projectors are relatively inexpensive. Finally, because they do not require a mirror and a rectilinear screen, front projectors produce a very sharp image, comparable to direct view.
Disadvantages: The biggest disadvantage to front projection technology is that you must install the projector in an environment that contains little or no ambient light when in use. Just like in a movie theater, even a little bit of light in the room can noticeably wash out the image. Another disadvantage is that installation of the projector and screen requires a considerable amount of floor space and planning. For this reason, front projectors are best suited to dedicated home theaters or at least dedicated recreational rooms.
Now let's consider the 5 major types of display technology.
Advantages: CRTs have three great advantages. First they are relatively inexpensive. You can purchase a really good 50-60" CRT rear projection display for well under $2000. Second, CRTs provide the standard for contrast, color accuracy, and black levels. The blacks on a CRT are truly black. Third, CRTs are quite bright, so you can easily watch them in rooms with lots of ambient light. In fact, according to many objective measurements, CRTs still offer the best picture possible.
Disadvantages: The image that CRT technology provides is not as sharp as that provided by most fixed-pixel displays. This is for two reasons. First, CRTs are an analog display technology and, all else being equal, a fixed-pixel display will always appear sharper than an analog image. Second, CRTs produce color by aligning three separate red, green, and blue guns, and it is really difficult to get the alignment perfect. Direct-view CRTs are also enormously heavy and limited in size. The biggest are 36" diagonal and they weigh from 200-250 lbs. Finally, CRTs are subject to burn-in. If you allow a fixed image to stay on the screen for a long period of time, it can leave a visible and semi-permanent after image. This is particularly a problem with wide screen rear projection CRTs because so much broadcast material is still not wide screen. In order to produce the correct aspect ratio, a wide screen set would have to display the image within vertical bars on the right and left sides. Because these bars tend to burn in, manufactures provide various stretch modes (some better than others) whereby a 4x3 image is painted across an entire 16x9 screen. This avoids burn in, but it creates noticeable geometric distortion.
In the last couple years, high-quality CRTs have almost completely disappeared from the marketplace. Consumers have just fallen in love with flat panel displays.
Advantages: LCDs produce a very bright, very sharp image. Also, LCD flat panels are quite thin and light weight. Finally LCDs are immune from burn-in.
Disadvantages: LCD flat panels are relatively expensive. As of this writing they are the most expensive, inch-for-inch, display technology around, though prices are trending downwards. LCD projectors, both front and rear, have traditionally suffered from poor black levels and a visible pixel structure, known as the screen door effect. Technological advances in the last couple of years have greatly reduced, though not entirely eliminated, these problems.
Advantages: Plasmas, like LCDs, produce a very bright, very sharp picture. Unlike LCDs, plasmas are only available in direct view models (there are no projection plasmas). Although heavier than LCDs, they are lighter than CRTs of an equivalent size. Finally, though their black levels are not a good as CRTs, plasma displays can provide really exceptional ANSI contrast.
Disadvantages: Though not as expensive as LCDs of 42" or higher, plasmas can get quite pricey. However, like all consumer electronics the cost is steadily trending downwards. Like CRTs, plasmas are subject to burn-in. Because they do not come in projection models, the size is limited to about 50". Sizes larger than this increase in price exponentially.
Advantages: Like LCDs, DLP is immune from burn-in. However, DLPs are generally much better than LCDs at providing high contrast with deep blacks (though not as deep as CRTs) and they have less of a problem with the screen door effect. Unlike LCDs, DLPs are only available as front and rear projection models (there are no direct view DLP displays).
Disadvantages: Because they are only available in projection sets, rear projection DLPs (which are far more common than front projection models), produce an image that is not as sharp as direct view LCDs or plasmas. Also, because all affordable DLPs contain only one chip, they rely upon a rotating color wheel to provide color. This can produce an artifact commonly referred to as the "rainbow effect (RBE)." This is less of an issue now than it was when DLPs first came out, but some people (a minority) still claim to experience severe viewing fatigue from watching DLPs because of the color separation artifacts created by the color wheel. The vast majority of people won't be affected by this. I can't see this at all without looking at test patterns and shaking my head back and forth.
Advantages: LCoS produces a very smooth, film-like image with the least amount of screen door effect of the fixed-pixel technologies. It is theoretically capable of very high contrast and good black levels, though we are only now starting to see models that realize this capacity.
Disadvantages: LCoS projectors tend to be more expensive than LCDs or DLPs. Perhaps the biggest disadvantage of LCoS is purely practical. Companies have had a lot of trouble mass producing LCoS chips, which has limited their effect on the marketplace. Lately though Sony seems to have solved this problem with their SXRD technology (SXRD is just Sony's proprietary name for LCoS). The biggest problem with LCoS display devices really had nothing to do with the technology itself, but rather on how the manufacturers have decided to implement it. Some very popular LCoS displays exhibit exaggerated color, sometimes extremely exaggerated color. Some consumers prefer this, but some recent LCoS displays have taken this trend to the point that the colors begin to look cartoonish and very unnatural, at least to me.
Conclusions
So returning to the original question, "what's the best type of display?", as we've seen with the 3 display types and 5 display technologies, how you answer this question depends on what you want.