You forgot to mention the normal DE filter. Pentair FNS Plus DE filter is the best as far as I am concerned.

There is a short comparison over at the TFP site that describes the differences without bias.

I use a cartridge, used to have DE, and am quite happy, but each owner, combined with their circumstances, will drive the answer in different directions.

Here is a link that might be useful: Pool Filter comparison

Depends on the pool, the surrounding, and who is taking care of it.

DE seems favored in most places for the cleaning, and high flow rates.
In very dirty areas I would recommend Sand.
If you are on a small lot with no place to backwash then I would recommend Cartridge.

I guess I should have told you a little about the pool. We have a 2.5 acres of open space and the pool will be 1000 sq ft with a 7 ft spa. We are in the category of pool builders who have to make something out of nothing. The yard is relatively flat so we will be using the dirt to build up a burm on the far side of the pool and a 15.5' Dolphin slide will be installed. My husband will be the one who will be dealing with filter system. There will be plenty of place to backwash.

What we have, and I highly recommend it, is a sand filter, but instead of using sand, we use Zeobrite.

Among the three choices of DE, cartridge, and sand (using pool filter sand), they're listed in order (best to worst) of filtering ability.

However, when you use Zeobrite in a sand filter instead of sand, you have the filtering capability of DE, but the ease of maintenance of sand.

I have a Pentair Quad 80 and Intelliflo. I use Fibre Clear instead of DE. They say that it polishes the water and it really does. My water is absolutely crystal clear. The pump runs at such a low speed that I can not even hear if it is on and only uses 120 watts. I am going to add the Pentair separation tank so it is easier to backwash and not waste water.