I have the Onkyo NR717 with all the modes you list. I did a HUGE amount of test/listening on all modes, and I will qualify my answer by saying you should go with what sounds best to YOUR ears in YOUR listening room.
For TV I like the Neo:6 and it is my default listening mode. It splits 2 channel sound from the TV into 5.1, and it does a pretty good job, I think. I get very clear dialog from the center channel, and I think it has a better bass than the DD.
If you play movies, the AVR will detect and play the codec, so if the movie was done in DD you will hear DD, and it sounds real good to my ears.
The DTS mode sounds pretty good with DVD/Blu-ray that uses that format. My 717 knows when I pop in a Blu-ray with DTS and uses it. You can change it to another mode, but the DTS mode sounds pretty good with clear dialog from the center channel at normal dialog listening levels, then when the explosions and music get going especially near reference listening levels, the DTS mode turns them up about a third louder than the dialog.
For music, there is a "Pure Audio" mode which takes out all the electronic enhancements, and I think it makes all the sounds more brilliant. For music I like it better than the Neo:6 because it is the most accurate sound.
For music, you might also try "all speakers stereo." If you have speakers which match each other pretty well, you can get a really "punchy" musical sound, and in general it will play louder with a more pleasant sound (less compressed) than other modes.
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With regards to Mr. Kwickness' comments, in my Onk 717 there is an adjustment you can make in the setup menu (tools menu) that lets you hear the sounds with the big swings in dynamic level (good for music and and loud movies), a medium level, and a mode which will pretty much tame the large sound swings, for a very pleasant movie or TV listening experience. The latter mode is much less "in your face" and is preferred by my wife, who doesn't like the loud sounds.