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If I do pigtail the outlets to the main 12/3 line going around my
basement, where's the appropriate location for the pigtail connections
relative to the main line? Putting the pigtail connection above the
box, between two joists, and surrounded by fiberglass insulation
doesn't seem correct to me.

All connections have to be made *inside* the box to prevent sparks from
igniting any material inside the wall cavities. If you're going to pigtail,
it would be a good idea to use extra deep outlet boxes to accommodate the
wire nuts that will be needed.

Personally, I'd not bother for two reasons. If you decide to use some sort
of automated outlet or dimmer switch, you'll find they are much larger than
standard switches and outlets and you'll be facing a very tight fit if
you've got three wirenuts to account for.

Also, if you wire through the back of the outlets and one outlet dies, it
may take the rest of the string with them but that's not such a bad thing.
It will tell you with a high degree of certainty which outlet needs
attention: the one that's failed closest to the main breakers.

Also, it seems that the physical connection using the terminal screws
on the outlets would be stronger than the connection provided by a
wirenut. Thoughts on that??

I suspect some of the replies you got were assuming you'd use the "backstab"
connections on the outlets. IMHO, those are quick and dirty and far more
likely to fail than either wire nuts or terminal screws. I just repaired
one for a friend whose disposal first became intermittent and then failed
completely. The outlet that it drew power from had been badly "backstabbed"
and the connection became loose. As soon as she told me that there was
another kitchen outlet that was flaky, I realized what the problem was.

Whatever you decide, make sure your wire stripping is clean and without
nicks. The failure point for outlets and switches, at least in my
experience, always seems to be at the point where the copper was nicked by
the stripper.

--
Bobby G.