Excellent topic, my friend. I too want to hear what others have to say about this. I'm one who tends to use too much of the same all the time, I fear, but all my playing jobs have been on electronic organs, and often there aren't many really different registrations that work well for congregational singing.
The old stand-by 8-4-2 combination of flutes and diapasons usually works well for me, though it's often instructive to leave out one or more of these from time to time just to get an idea of how the overall tone color can be varied this way. I discovered the joy of the Quinte 2 2/3 a few years back when playing on an Allen ADC4000 for a few months. This clear but soft principal stop added just the right amount of faux reed color to my principal chorus when I wanted to beef it up without using a real reed. On my current church organ, I often use both the 2 2/3 and the 1 3/5 members of the swell flute chorus (which is coupled to the great) as this adds a rather sweet reed-like quality to the tone without adding a lot of volume. (This only works when these mutation stops are independent and rather soft. A typical analog organ with a unified swell flute may have mutations that are too loud to use this way.)
If playing an electronic that has separate audio systems for the swell and great, I'll always use the swell to great coupler and select a similar or supplementary set of stops on the swell. I would think it good to have the great and swell very similar in color and volume, especially in a divided installation where some in the congregation hear primarily one division or the other. Ideally, of course, with a good installation and good acoustics everyone hears the organ about equally.
As far as mixtures, I am generally in favor of them, as they add sparkle and clarity to the organ tone, enhancing the harmonic series, as has been pointed out elsewhere. If the congregation is large and singing well, the mixtures will help to carry the organ tone above the singing so that everyone can stay on pitch and in tempo.
However, some electronics, such as the poor little Galanti I'm currently playing on Sundays, have peculiar mixtures that may be either too shrill or too loud for frequent use. In our congregation, which contains quite a few older folks with hearing aids, some say that the upper frequencies of the organ hurt their ears or cause their hearing aids to distort. I think the problem is with this organ's mixtures, which sound too fluty to me, not enough real clarity, not enough of the highest overtones perhaps. By contrast, the Allen MDS45 we are working on the shop has mixtures that really seem to float right out of the speakers, bold and bright without being heavy or shrill. I wish all mixtures had that kind of quality!
So, my general formula for a hymn registration would be to draw the great division's flute and principal stops at 8-4-2 (though, as I said, sometimes leaving out one or two for variety), and also draw a similar set of stops in the swell, drawing the swell to great coupler and playing on the great. I might use a mixture on one or both divisions, might throw in a 2 2/3 stop from time to time, and would use an 8' reed when real power is required. For quieter hymns, I might use only 8's and 4's, and I've found it's actually OK to add the mixture to that combination without first adding the 2' stop, for another interesting variation.
Pedal stops for congregational singing need to be quite bold, as people enjoy the big bass, the physical sensation of the lowest frequencies hitting their bodies. I normally use a couple of full-throated 16' stops (not reeds except for really big hymns) plus a flute and a principal at 8'. If the organ has a good independent pedal division, I use the 4' stops, maybe even the pedal mixtures. Otherwise, I use the swell to pedal coupler, but usually not the great to pedal. This way, it is always possible to switch hands to the swell for a quick dynamic change and still have the pedal balance with the manual.
I'd love to hear what others do, especially those who play pipes.